Notes on Convex Sets, Polytopes, Polyhedra Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations

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Combinatorial Topology, Voronoi Diagrams and

Delaunay Triangulations

Jean Gallier

To cite this version:

Jean Gallier. Notes on Convex Sets, Polytopes, Polyhedra Combinatorial Topology, Voronoi Diagrams

and Delaunay Triangulations. [Research Report] RR-6379, INRIA. 2007, pp.191. �inria-00193831v3�

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Thème BIO

Notes on Convex Sets, Polytopes, Polyhedra

Combinatorial Topology, Voronoi Diagrams and

Delaunay Triangulations

Jean Gallier

N° 6379

Delaunay Triangulations

Jean Gallier

∗

ThèmeBIOSystèmesbiologiques ProjetAs lepios

Rapportdere her he n°6379De ember2007191pages

Abstra t: Somebasi mathemati altoolssu has onvexsets, polytopesand ombinato-rialtopology,areused quiteheavilyinapplied eldssu hasgeometri modeling, meshing, omputervision,medi alimagingandroboti s. Thisreportmaybeviewedasatutorialand aset of notes on onvexsets, polytopes, polyhedra, ombinatorial topology, Voronoi Dia-gramsandDelaunayTriangulations. Itisintended forabroadaudien eofmathemati ally in linedreaders.

Oneof my (selsh!) motivations in writingthese notes wasto understand the on ept ofshelling andhowitisusedto provethefamousEuler-Poin aréformula(Poin aré,1899) andthemorere entUpperBoundTheorem (M Mullen,1970)forpolytopes. Anotherofmy motivationswastogivea orre ta ountofDelaunaytriangulationsandVoronoidiagrams intermsof(dire tandinverse)stereographi proje tionsontoasphereandproverigorously that the proje tive map that sends the (proje tive) sphere to the (proje tive) paraboloid works orre tly, that is, maps theDelaunaytriangulation andVoronoi diagram w.r.t. the liftingontothespheretotheDelaunaydiagramandVoronoidiagramsw.r.t. thetraditional liftingontotheparaboloid. Here,theproblemisthatthismapisonlywelldened(total)in proje tivespa eandwearefor edto denethe notionof onvexpolyhedronin proje tive spa e.

Itturnsoutthatinordertoa hieve(evenpartially)theabovegoals,Ifoundthatitwas ne essarytoin ludequiteabitofba kgroundmaterialon onvexsets,polytopes,polyhedra and proje tivespa es. I havein luded a rather thorough treatment of the equivalen e of

V

-polytopesand

H

-polytopesandalsooftheequivalen eof

V

-polyhedraand

H

-polyhedra, whi h isabit harder. Inparti ular,theFourier-Motzkin elimination method(a versionof Gaussianeliminationforinequalities)isdis ussedinsomedetail. Ialsohadtoin ludesome

∗

Onsabbati alfromtheUniversityofPennsylvania.DepartmentofComputerandInformationS ien e. Philadelphia,PA19104,USA

material on proje tive spa es, proje tive maps and polar duality w.r.t. a nondegenerate quadri in orderto dene asuitablenotionof proje tivepolyhedron basedon ones. To thebestofourknowledge,thisnotionofproje tivepolyhedronisnew. Wealsobelievethat someofourproofsestablishingtheequivalen eof

V

-polyhedraand

H

-polyhedra arenew. Key-words: Convex sets, polytopes, polyhedra, shellings, ombinatorial topology, Voronoidiagrams,Delaunaytriangulations.

Voronoi et les Triangulations de Delaunay

Résumé: Desoutilsmathématiquesdebasetelsquelesensembles onvexes,lespolytopes et la topologie ombinatoire, sont beau oup utilisés en modélisation géométrique, vision, maillage,imageriemédi aleetrobotique. Cerapportpeutêtre onsidéré ommeuntutorial etunensembledenotessurlesensembles onvexes,lespolytopes,lespolyhèdres,latopologie ombinatoire, les diagramesde Voronoi et lestriangulations deDelaunay. Ilest destiné à unelargeaudien eayantunein linationmathématique.

Unedemesmotivations(egoïste!) enrédigeant esnotesétaitde omprendrele on ept d'éeuillage et de voir omment il est utilisé pour démontrer la élèbre formule d'Euler-Poin aré (Poin aré, 1899) et le plus ré ent Théorème de la borne supérieure (M Mullen, 1970) pour les polytopes. Une autre de mes motivations était de donner un traitement  orre t destriangulations de Delaunay et des diagramesde Voronoi àpartirdes proje -tionsstéréographiques(dire teset inverses)surune sphère. Jeprouverigoureusementque l'appli ationquitransformelasphère(proje tive) enunparaboloïde(proje tif)aun om-portement orre t, 'est-à-dire,fait orrespondrelatriangulationdeDelaunayetlediagrame de Voronoi parrapport au relèvement sur la sphère àla triangulation deDelaunay et au diagramedeVoronoiparrapportaurelèvementtraditionelsurleparaboloïde. Leproblème est que ette orresponden e n'estbien dénie (totale) quedans l'espa e proje tif et nous sommesdon obligésdedénirlanotiondepolyhèdre onvexedansl'espa eproje tif.

Il s'avèreque pouratteindre nos obje tifs (même partiellement),j'ai trouvéné essaire d'in lureune revue de ertainesnotionsde base telles que lesensembles onvexes, les po-lytopes, lespolyhèdres et lesespa es proje tifs. J'ai in lu un traitementassez détaillé de l'équivalen edes

V

-polytopesetdes

H

-polytopesainsiquel'équivalen edes

V

-polyhèdreset des

H

-polyhèdres,quiestunpeuplusdi ile. En parti ulier,laméthoded'élimination de Fourier-Motzkin (uneversiondelaméthodedel'éliminationGaussiennepourlesinégalités) est traitée en détail. J'ai du également in lure un traitement des espa es proje tifs, des appli ationsproje tiveset deladualitépolaireparrapportàunequadriquenondégénérée andedénirunenotion onvenabledepolyhèdreproje tif reposantsurles ones. Ilnous sembleque ettenotionde polyhèdre proje tif est originale. Nouspensonségalementque ertainesdes preuves établissant l'équivalen e des

V

-polyhèdres et des

H

-polyhèdres sont originales.

Mots- lés : Ensembles onvexes,polytopes,polyhèdres,eeuillages,topologie ombina-toires,diagramesdeVoronoi,triangulationsdeDelaunay.

Contents

1 Introdu tion 7

1.1 MotivationsandGoals . . . 7

2 Basi Properties of Convex Sets 11 2.1 ConvexSets . . . 11

2.2 Carathéodory'sTheorem. . . 12

2.3 Verti es,Extremal PointsandKreinandMilman's Theorem. . . 15

2.4 Radon'sandHelly'sTheoremsandCenterpoints . . . 18

3 Separation and Supporting Hyperplanes 23 3.1 SeparationTheoremsandFarkasLemma. . . 23

3.2 SupportingHyperplanesandMinkowski'sProposition . . . 35

3.3 PolarityandDuality . . . 36

4 Polyhedra and Polytopes 43 4.1 Polyhedra,

H

-Polytopesand

V

-Polytopes . . . 43

4.2 TheEquivalen eof

H

-Polytopesand

V

-Polytopes. . . 53

4.3 TheEquivalen eof

H

-Polyhedraand

V

-Polyhedra . . . 54

4.4 Fourier-MotzkinElimination andCones . . . 60

5 Proje tive Spa esand Polyhedra,Polar Duality 69 5.1 Proje tiveSpa es . . . 69

5.2 Proje tivePolyhedra . . . 76

5.3 TangentSpa esofHypersurfa es . . . 83

5.4 Quadri s(Ane,Proje tive)andPolarDuality . . . 88

6 Basi s ofCombinatorialTopology 97 6.1 Simpli ialandPolyhedral Complexes . . . 97

7 Shellingsand the Euler-Poin aréFormula 113

7.1 Shellings . . . 113

7.2 TheEuler-Poin aréFormulaforPolytopes . . . 123

7.3 Dehn-SommervilleEquationsforSimpli ialPolytopes . . . 126

7.4 TheUpperBoundTheorem . . . 133

8 Diri hletVoronoi Diagrams 141 8.1 Diri hletVoronoiDiagrams . . . 141

8.2 Triangulations . . . 148

8.3 DelaunayTriangulations . . . 152

8.4 DelaunayTriangulationsandConvexHulls . . . 154

8.5 Stereographi Proje tionandtheSpa eof Spheres . . . 157

8.6 Stereographi Proje tionandDelaunayPolytopes. . . 175

Chapter 1

Introdu tion

1.1 Motivations and Goals

ForthepasteightyearsorsoIhavebeentea hingagraduate oursewhosemain goalisto exposestudentsto somefundamental on eptsofgeometry,keepingin mind their appli a-tionstogeometri modeling,meshing, omputervision,medi alimaging,roboti s,et . The audien ehasbeenprimarily omputers ien estudents but afair numberof mathemati s studentsand alsostudentsfrom other engineeringdis iplines(su h asEle tri al,Systems, Me hani alandBioengineering)havebeenattendingmy lasses. Inthepastthreeyears,I havebeenfo using moreon onvexity, polytopes and ombinatorial topology, as on epts andtoolsfromtheseareashavebeenusedin reasinglyinmeshingandalsoin omputational biologyandmedi alimaging. Oneofmy(selsh!) motivationswasto understandthe on- ept ofshelling and how itis usedto provethe famousEuler-Poin aréformula (Poin aré, 1899)andthemorere entUpperBoundTheorem (M Mullen,1970)forpolytopes. Another ofmy motivationswasto givea orre t a ountof Delaunay triangulationsandVoronoi diagramsintermsof(dire tandinverse)stereographi proje tionsontoasphereandprove rigorously that the proje tive map that sends the (proje tive) sphere to the (proje tive) paraboloidworks orre tly,that is, mapstheDelaunaytriangulationandVoronoidiagram w.r.t. theliftingonto thespheretotheDelaunaytriangulationandVoronoidiagramw.r.t. theliftingontothe paraboloid. Moreover,theproje tionsof these polyhedra onto the hy-perplane

x

d+1

= 0

,fromthesphereorfromtheparaboloid,areidenti al. Here,theproblem isthat thismap isonlywelldened(total) in proje tivespa e andweare for edto dene thenotionof onvexpolyhedronin proje tivespa e.

Itturnsoutthatinordertoa hieve(evenpartially)theabovegoals,Ifoundthatitwas ne essarytoin ludequiteabitofba kgroundmaterialon onvexsets,polytopes,polyhedra and proje tivespa es. I havein luded a rather thorough treatment of the equivalen e of

V

-polytopesand

H

-polytopesandalsooftheequivalen eof

V

-polyhedraand

H

-polyhedra, whi h isabit harder. Inparti ular,theFourier-Motzkin elimination method(a versionof

Gaussianeliminationforinequalities)isdis ussedinsomedetail. Ialsohadtoin ludesome material on proje tive spa es, proje tive maps and polar duality w.r.t. a nondegenerate quadri ,inordertodeneasuitablenotionofproje tivepolyhedronbasedon ones. This notionturnedouttobeindispensibletogivea orre ttreatmentoftheDelaunayandVoronoi omplexesusinginversestereographi proje tionontoasphereandtoproverigorouslythat thewell knownproje tivemapbetweenthesphereand theparaboloidmaps theDelaunay triangulationand theVoronoidiagram w.r.t. the sphereto themoretraditionalDelaunay triangulationandVoronoidiagramw.r.t. theparaboloid. Tothebestofourknowledge,this notionofproje tivepolyhedronisnew. Wealsobelievethatsomeofourproofsestablishing theequivalen eof

V

-polyhedra and

H

-polyhedraarenew.

Chapter 6 on ombinatorial topology is hardly original. However,most texts overing thismaterialareeitheroldfashionortooadvan ed. Yet,thismaterialisusedextensivelyin meshingandgeometri modeling. Wetriedtogivearatherintuitiveyetrigorousexposition. Wede idedtointrodu etheterminology ombinatorial manifold,anotionusuallyreferred toastriangulatedmanifold.

Are urringthemeinthesenotesisthepro essof oni ation (algebrai ally, homoge-nization),thatis, forminga onefrom somegeometri obje t. Indeed, oni ation turns anobje tinto aset oflines, and sin elinesplaythe roleof pointsin proje tivegeometry,  oni ation(homogenization)isthewaytoproje tivizegeometri aneobje ts. Then, these (ane) obje tsappear as oni se tions of onesby hyperplanes, just thewaythe lassi al oni s(ellipse,hyperbola,parabola)appearas oni se tions.

Itisworthwarningourreadersthat onvexityandpolytopetheoryisde eptivelysimple. Thisisasubje twheremostintuitivepropositionsfailassoonasthedimensionofthespa e isgreaterthan

3

(denitely

4

),be auseourhumanintuitionisnotverygoodindimension greaterthan

3

. Furthermore,rigorousproofsofseeminglyverysimplefa tsareoftenquite ompli atedandmayrequiresophisti atedtools(forexample,shellings,fora orre tproofof theEuler-Poin aréformula). Nevertheless,readersareurgedtostrenghtentheirgeometri intuition;theyshouldjustbeveryvigilant! Thisisanother asewhereTate'sfamoussaying ismorethanpertinent: Reasongeometri ally,provealgebrai ally.

At rst, these notes were meant asa omplement to Chapter 3(Properties of Convex Sets: A Glimpse)ofmy book (Geometri Methodsand Appli ations, [20℄). However,they turnoutto overmu hmorematerial. Forthereader's onvenien e,Ihavein ludedChapter 3ofmybookaspartofChapter2ofthesenotes. Ialsoassumesomefamiliaritywithane geometry. Thereadermaywishtoreviewthebasi sofanegeometry. These anbefound inanystandardgeometrytext(Chapter2ofGallier[20℄ oversmorethanneededforthese notes).

Mostofthematerialon onvexsetsistakenfromBerger[6℄(GeometryII).Otherrelevant sour es in lude Ziegler [43℄, Grünbaum [24℄ Valentine [41℄, Barvinok [3℄, Ro kafellar [32℄, Bourbaki(Topologi alVe torSpa es)[9℄andLax[26℄,thelastfourdealingwithanespa es of innitedimension. As to polytopesand polyhedra, the lassi referen eis Grünbaum [24℄. Othergoodreferen esin ludeZiegler[43℄,Ewald[18℄,Cromwell[14℄andThomas[38℄.

There entbookbyThomas ontainsanex ellentand easygoingpresentation of poly-topetheory. This book also givesan introdu tionto the theoryof triangulationsof point ongurations,in luding thedenition of se ondarypolytopesand statepolytopes,whi h happentoplayarolein ertainareasofbiology. Forthis,aqui kbut verye ient presen-tationof Gröbnerbases isprovided. Wehighly re ommendThomas'sbook[38℄ asfurther reading. Itis alsoan ex ellentpreparationforthe moreadvan edbook bySturmfels [37℄. However,inouropinion,thebibleonpolytopetheoryiswithoutany ontest,Ziegler[43℄, amasterlyandbeautifulpie eofmathemati s. Infa t,ourChapter7isheavilyinspiredby Chapter 8ofZiegler. However,thepa e of Ziegler'sbook is quitebrisk andwehopethat ourmorepedestriana ountwillinspirereaderstogoba kandreadthemasters.

Inanottoodistantfuture,Iwouldliketowriteabout onstrainedDelaunay triangula-tions,aformidabletopi ,pleasebepatient!

I wish to thank Mar elo Siqueira for at hing many typos and mistakes and for his manyhelpfulsuggestionsregardingthepresentation. Atleastathirdofthismanus riptwas writtenwhileIwasonsabbati alatINRIA, SophiaAntipolis,intheAs lepiosProje t. My deepest thanksto Ni holasAya heand his olleagues(espe ially XavierPenne andHervé Delingette)forinvitingme tospend awonderfulandveryprodu tiveyear andformaking mefeelperfe tly athomewithintheAs lepiosProje t.

Chapter 2

Basi Properties of Convex Sets

2.1 Convex Sets

Convexsetsplayaveryimportantroleingeometry. Inthis hapterwestateandprovesome ofthe lassi s of onvexanegeometry: Carathéodory'stheorem, Radon'stheorem,and Helly's theorem. These theorems share theproperty that theyareeasy to state,but they aredeep,andtheirproof,althoughrathershort,requiresalotof reativity.

Given an ane spa e

E

, re all that a subset

V

of

E

is onvex if for any two points

a, b ∈ V

, wehave

c ∈ V

foreverypoint

c = (1 − λ)a + λb

,with

0 ≤ λ ≤ 1

(

λ ∈ R

). Given anytwopoints

a, b

,the notation

[a, b]

isoftenused to denote theline segmentbetween

a

and

b

,that is,

[a, b] = {c ∈ E | c = (1 − λ)a + λb, 0 ≤ λ ≤ 1},

andthusaset

V

is onvexif

[a, b] ⊆ V

foranytwopoints

a, b ∈ V

(

a = b

isallowed). The emptyset istrivially onvex,everyone-pointset

{a}

is onvex,and theentire anespa e

E

isof ourse onvex.

Itisobviousthattheinterse tionofanyfamily(niteorinnite)of onvexsetsis onvex. Then, given any (nonempty) subset

S

of

E

, there is a smallest onvex set ontaining

S

denotedby

C(S)

or

conv(S)

and alledthe onvexhull of

S

(namely,theinterse tionofall onvexsets ontaining

S

). The ane hull of a subset,

S

, of

E

is the smallest ane set ontaining

S

anditwill bedenotedby

hSi

or

aff(S)

.

Agoodunderstandingofwhat

C(S)

is,andgoodmethodsfor omputingit,areessential. First,wehavethefollowingsimplebut ru iallemma:

Lemma2.1 Given an ane spa e

E,

−

→

E , +



, for any family

(a

i

)

i∈I

of points in

E

, the set

V

of onvex ombinations

P

i∈I

λ

i

a

i

(where

P

i∈I

λ

i

= 1

and

λ

i

≥ 0

) isthe onvexhull of

(a

i

)

i∈I

.

Proof. If

(a

i

)

i∈I

isempty,then

V = ∅

,be auseofthe ondition

P

i∈I

λ

i

= 1

. Asinthe ase ofane ombinations,itis easilyshown byindu tion thatany onvex ombination anbe obtainedby omputing onvex ombinationsoftwopointsat atime. Asa onsequen e,if

(a

i

)

i∈I

isnonempty,thenthesmallest onvexsubspa e ontaining

(a

i

)

i∈I

must ontainthe set

V

of all onvex ombinations

P

i∈I

λ

i

a

i

. Thus, it is enough to show that

V

is losed under onvex ombinations,whi hisimmediatelyveried.

InviewofLemma2.1,itisobviousthatanyanesubspa eof

E

is onvex. Convexsets alsoarisein termsofhyperplanes. Givenahyperplane

H

, if

f : E → R

isanynon onstant aneformdening

H

(i.e.,

H = Ker f

),we andenethetwosubsets

H

+

(f ) = {a ∈ E | f (a) ≥ 0}

and

H

−

(f ) = {a ∈ E | f (a) ≤ 0},

alled( losed)half-spa es asso iatedwith

f

.

Observethat if

λ > 0

,then

H

+

(λf ) = H

+

(f )

,butif

λ < 0

,then

H

+

(λf ) = H

−

(f )

,and similarlyfor

H

−

(λf )

. However,theset

{H

+

(f ), H

−

(f )}

depends only on the hyperplane

H

, and the hoi e of a spe i

f

dening

H

amounts to the hoi e of oneof thetwo half-spa es. Forthis reason, we will also say that

H

+

(f )

and

H

−

(f )

are the losed half-spa es asso iated with

H

. Clearly,

H

+

(f ) ∪ H

−

(f ) = E

and

H

+

(f ) ∩ H

−

(f ) = H

. It is immediately veried that

H

+

(f )

and

H

−

(f )

are onvex. Bounded onvexsets arisingasthe interse tion of anite familyof half-spa esasso iated withhyperplanesplayamajorrolein onvexgeometryandtopology(theyare alled onvex polytopes).

ItisnaturaltowonderwhetherLemma2.1 anbesharpenedin twodire tions: (1)Isit possibletohaveaxedboundonthenumberofpointsinvolvedinthe onvex ombinations? (2)Isitne essaryto onsider onvex ombinationsofallpoints,orisitpossibleto onsider onlyasubsetwithspe ialproperties?

Theanswerisyesinboth ases. In ase1,assumingthattheanespa e

E

hasdimension

m

,Carathéodory'stheoremassertsthatitisenoughto onsider onvex ombinationsof

m+1

points. Forexample, in the plane

A

2

, the onvexhullof aset

S

of pointsis the unionof alltriangles (interiorpoints in luded) with verti esin

S

. In ase 2,the theorem of Krein andMilmanassertsthata onvexsetthatisalso ompa tisthe onvexhullofitsextremal points(givena onvexset

S

,apoint

a ∈ S

isextremalif

S − {a}

isalso onvex,seeBerger [6℄ orLang[25℄). Next,weproveCarathéodory'stheorem.

2.2 Carathéodory's Theorem

Theproof ofCarathéodory'stheoremis reallybeautiful. It pro eedsby ontradi tionand usesaminimalityargument.

Theorem2.2 Given any ane spa e

E

of dimension

m

, for any (nonvoid) family

S =

(a

i

)

i∈L

in

E

,the onvexhull

C(S)

of

S

isequaltothesetof onvex ombinationsoffamilies of

m + 1

points of

S

. Proof. ByLemma2.1,

C(S) =

X

i∈I

λ

i

a

i

| a

i

∈ S,

X

i∈I

λ

i

= 1, λ

i

≥ 0, I ⊆ L, I

nite



.

Wewouldliketoprovethat

C(S) =

X

i∈I

λ

i

a

i

| a

i

∈ S,

X

i∈I

λ

i

= 1, λ

i

≥ 0, I ⊆ L, |I| = m + 1



.

Wepro eedby ontradi tion. Ifthetheoremisfalse,thereissomepoint

b ∈ C(S)

su hthat

b

anbeexpressed as a onvex ombination

b =

P

i∈I

λ

i

a

i

, where

I ⊆ L

is anite set of ardinality

|I| = q

with

q ≥ m + 2

,and

b

annotbeexpressed asany onvex ombination

b =

P

_j∈J

µ

j

a

j

of stri tly fewer than

q

points in

S

, that is, where

|J| < q

. Su h apoint

b ∈ C(S)

isa onvex ombination

b = λ

1

a

1

+ · · · + λ

q

a

q

,

where

λ

1

+ · · · + λ

q

= 1

and

λ

i

> 0 (1 ≤ i ≤ q

). Weshall provethat

b

anbewritten asa onvex ombination of

q − 1

ofthe

a

i

. Pi kanyorigin

O

in

E

. Sin ethere are

q > m + 1

points

a

1

, . . . , a

q

,thesepointsareanelydependent,andbyLemma2.6.5fromGallier[20℄, thereisafamily

(µ

1

, . . . , µ

q

)

alls alarsnotallnull,su hthat

µ

1

+ · · · + µ

q

= 0

and

q

X

i=1

µ

i

Oa

i

= 0.

Considertheset

T ⊆ R

denedby

T = {t ∈ R | λ

i

+ tµ

i

≥ 0, µ

i

6= 0, 1 ≤ i ≤ q}.

Theset

T

is nonempty, sin eit ontains

0

. Sin e

P

q

i=1

µ

i

= 0

and the

µ

i

arenotall null, therearesome

µ

h

, µ

k

su hthat

µ

h

< 0

and

µ

k

> 0

,whi himpliesthat

T = [α, β]

, where

α = max

1≤i≤q

{−λ

i

/µ

i

| µ

i

> 0}

and

β = min

1≤i≤q

{−λ

i

/µ

i

| µ

i

< 0}

(

T

istheinterse tion ofthe losedhalf-spa es

{t ∈ R | λ

i

+ tµ

i

≥ 0, µ

i

6= 0}

). Observethat

α < 0 < β

,sin e

λ

i

> 0

forall

i = 1, . . . , q

.

We laimthatthereissome

j (1 ≤ j ≤ q)

su hthat

Indeed,sin e

α = max

1≤i≤q

{−λ

i

/µ

i

| µ

i

> 0},

asthesetontherighthandsideisnite,themaximumisa hievedandthereissomeindex

j

sothat

α = −λ

j

/µ

j

. If

j

issomeindexsu hthat

λ

j

+ αµ

j

= 0

,sin e

P

q

i=1

µ

i

Oa

i

= 0

,we have

b =

q

X

i=1

λ

i

a

i

= O +

q

X

i=1

λ

i

Oa

i

+ 0,

= O +

q

X

i=1

λ

i

Oa

i

+ α



q

X

i=1

µ

i

Oa

i



,

= O +

q

X

i=1

(λ

i

+ αµ

i

)Oa

i

,

=

q

X

i=1

(λ

i

+ αµ

i

)a

i

,

=

q

X

i=1, i6=j

(λ

i

+ αµ

i

)a

i

,

sin e

λ

j

+ αµ

j

= 0

. Sin e

P

q

i=1

µ

i

= 0

,

P

q

i=1

λ

i

= 1

,and

λ

j

+ αµ

j

= 0

,wehave

q

X

i=1, i6=j

λ

i

+ αµ

i

= 1,

andsin e

λ

i

+ αµ

i

≥ 0

for

i = 1, . . . , q

,theaboveshowsthat

b

anbeexpressedasa onvex ombinationof

q − 1

pointsfrom

S

. However,this ontradi tstheassumptionthat

b

annot beexpressedasa onvex ombinationofstri tlyfewerthan

q

pointsfrom

S

,andthetheorem isproved.

If

S

is a nite (of innite) set of points in the ane plane

A

2

, Theorem 2.2 onrms ourintuition that

C(S)

is the unionof triangles (in ludinginteriorpoints) whose verti es belong to

S

. Similarly,the onvexhullofaset

S

of pointsin

A

3

istheunionoftetrahedra (in ludinginteriorpoints)whoseverti esbelongto

S

. Wegetthefeelingthattriangulations playa ru ialrole,whi hisof oursetrue!

Nowthatwehavegivenananswertotherstquestionposedat theendofSe tion 2.1 wegiveananswertothese ondquestion.

2.3 Verti es, Extremal Points and Krein and Milman's Theorem

First,wedenethenotionsofseparationandofseparatinghyperplanes. Forthis,re allthe denitionofthe losed(oropen)halfspa esdeterminedbyahyperplane.

Given a hyperplane

H

, if

f : E → R

is any non onstant ane form dening

H

(i.e.,

H = Ker f

),wedenethe losedhalf-spa esasso iatedwith

f

by

H

+

(f ) = {a ∈ E | f (a) ≥ 0},

H

−

(f ) = {a ∈ E | f (a) ≤ 0}.

Observethat if

λ > 0

,then

H

+

(λf ) = H

+

(f )

,butif

λ < 0

,then

H

+

(λf ) = H

−

(f )

,and similarlyfor

H

−

(λf )

.

Thus, theset

{H

+

(f ), H

−

(f )}

dependsonlyon thehyperplane,

H

,and the hoi eof a spe i

f

dening

H

amountstothe hoi eofoneofthetwohalf-spa es.

Wealsodenetheopen halfspa esasso iatedwith

f

asthetwosets

◦

H

+

(f ) = {a ∈ E | f (a) > 0},

◦

H

−

(f ) = {a ∈ E | f (a) < 0}.

Theset

{

◦

H

+

(f ),

◦

H

−

(f )}

onlydependsonthehyperplane

H

. Clearly,wehave

◦

H

+

(f ) =

H

+

(f ) − H

and

◦

H

−

(f ) = H

−

(f ) − H

.

Denition2.1 Givenan anespa e,

X

,and twononemptysubsets,

A

and

B

, of

X

, we saythat ahyperplane

H

separates(resp. stri tly separates)

A

and

B

if

A

isin oneand

B

isintheotherofthetwohalfspa es(resp. openhalfspa es)determinedby

H

.

Thespe ial aseofseparationwhere

A

is onvexand

B = {a}

,forsomepoint,

a

,in

A

, isofparti ularimportan e.

Denition2.2 Let

X

be an ane spa e and let

A

be any nonempty subset of

X

. A supporting hyperplane of

A

is any hyperplane,

H

, ontaining some point,

a

, of

A

, and separating

{a}

and

A

. Wesaythat

H

is asupportinghyperplane of

A

at

a

.

Observe that if

H

is asupporting hyperplane of

A

at

a

, then we must have

a ∈ ∂A

. Otherwise,there wouldbe someopenball

B(a, ǫ)

of enter

a

ontainedin

A

and so there would be pointsof

A

(in

B(a, ǫ)

) in both half-spa esdetermined by

H

, ontradi tingthe fa tthat

H

isasupportinghyperplaneof

A

at

a

. Furthermore,

H ∩

◦

Figure2.1: Examplesofsupportinghyperplanes

Oneshouldexperimentwithvariouspi turesandrealizethatsupportinghyperplanesat apointmaynotexist (forexample,if

A

isnot onvex), maynotbeunique,andmayhave severaldistin tsupportingpoints!

Next,weneedtodenevarioustypesofboundarypointsof losed onvexsets.

Denition2.3 Let

X

be an ane spa e of dimension

d

. For any nonempty losed and onvexsubset,

A

, of dimension

d

, apoint

a ∈ ∂A

hasorder

k(a)

if theinterse tion of all thesupportinghyperplanesof

A

at

a

is ananesubspa eof dimension

k(a)

. Wesaythat

a ∈ ∂A

isavertex if

k(a) = 0

;wesaythat

a

issmooth if

k(a) = d − 1

,i.e.,ifthesupporting hyperplaneat

a

isunique.

A vertex is a boundary point,

a

, su h that there are

d

independent supporting hy-perplanes at

a

. A

d

-simplex has boundary points of order

0, 1, . . . , d − 1

. The following propositionisshowninBerger[6℄(Proposition11.6.2):

Proposition 2.3 The setof verti es ofa losedand onvexsubsetis ountable. Anotherimportant on eptisthatof anextremalpoint.

Denition2.4 Let

X

be an ane spa e. For any nonempty onvex subset,

A

, a point

a ∈ ∂A

isextremal (orextreme)if

A{a}

isstill onvex.

It is fairly obviousthat apoint

a ∈ ∂A

isextremal ifit doesnot belong to any losed nontriviallinesegment

[x, y] ⊆ A

(

x 6= y

).

Observethatavertexisextremal,butthe onverseisfalse. Forexample,in Figure2.2, allthepointsonthear ofparabola,in luding

v

1

and

v

2

,areextremepoints. However,only

v

1

v

2

Figure2.2: Examplesofverti esandextremepoints

v

1

and

v

2

areverti es. Also,if

dim X ≥ 3

,theset ofextremal pointsofa ompa t onvex maynotbe losed.

A tually,itisnotatallobviousthatanonempty ompa t onvexsetpossessesextremal points. Infa t,astrongerresultsholds (Kreinand Milman'stheorem). Inpreparationfor theproof of this importanttheorem, observethat any ompa t (nontrivial) intervalof

A

1

hastwoextremalpoints,itstwoendpoints. Weneedthefollowinglemma:

Lemma2.4 Let

X

be an ane spa e of dimension

n

, and let

A

be a nonempty ompa t and onvexset. Then,

A = C(∂A)

,i.e.,

A

isequaltothe onvexhull ofitsboundary. Proof. Pi k any

a

in

A

, and onsider any line,

D

, through

a

. Then,

D ∩ A

is losed and onvex.However,sin e

A

is ompa t,itfollowsthat

D∩A

isa losedinterval

[u, v]

ontaining

a

,and

u, v ∈ ∂A

. Therefore,

a ∈ C(∂A)

,asdesired.

The followingimportanttheorem showsthat only extremalpointsmatter asfaras de-termininga ompa tand onvexsubsetfromitsboundary. TheproofofTheorem2.5makes use of aproposition due to Minkowski (Proposition 3.17) whi h will beprovedin Se tion 3.2.

Theorem2.5 (KreinandMilman, 1940) Let

X

bean anespa eof dimension

n

. Every ompa t and onvex nonempty subset,

A

, isequal to the onvexhull of its set of extremal points.

Proof. Denote thesetofextremalpointsof

A

by

Extrem(A)

. Wepro eedbyindu tionon

d = dim X

. When

d = 1

,the onvexand ompa tsubset

A

mustbea losedinterval

[u, v]

, or a single point. In either ases, the theorem holds trivially. Now, assume

d ≥ 2

, and assumethatthetheorem holdsfor

d − 1

. Itiseasilyveriedthat

foreverysupportinghyperplane

H

of

A

(su hhyperplanesexist,byMinkowski'sproposition (Proposition3.17)). Observethat Lemma2.4impliesthatifwe anprovethat

∂A ⊆ C(Extrem(A)),

then,sin e

A = C(∂A)

,wewillhaveestablishedthat

A = C(Extrem(A)).

Let

a ∈ ∂A

,andlet

H

beasupporting hyperplaneof

A

at

a

(whi hexists, byMinkowski's proposition). Now,

A ∩ H

is onvex and

H

has dimension

d − 1

, and by the indu tion hypothesis, wehave

A ∩ H = C(Extrem(A ∩ H)).

However,

C(Extrem(A ∩ H)) = C((Extrem(A)) ∩ H)

= C(Extrem(A)) ∩ H ⊆ C(Extrem(A)),

andso,

a ∈ A ∩ H ⊆ C(Extrem(A))

. Therefore,weprovedthat

∂A ⊆ C(Extrem(A)),

fromwhi hwededu ethat

A = C(Extrem(A))

,asexplainedearlier.

Remark: Observethat KreinandMilman's theorem impliesthat anynonempty ompa t and onvexset hasanonemptysubsetof extremal points. Thisis intuitivelyobvious, but hardtoprove! KreinandMilman'stheoremalsoappliestoinnitedimensionalanespa es, providedthattheyarelo ally onvex,seeValentine[41℄,Chapter11,Bourbaki[9℄,Chapter II,Barvinok[3℄,Chapter3,orLax[26℄,Chapter13.

We on ludethis hapterwiththreeother lassi sof onvexgeometry.

2.4 Radon's and Helly's Theorems and Centerpoints WebeginwithRadon'stheorem.

Theorem2.6 Givenanyanespa e

E

ofdimension

m

,for everysubset

X

of

E

,if

X

has atleast

m + 2

points, then thereisapartition of

X

into twononemptydisjoint subsets

X

1

and

X

2

su hthat the onvexhullsof

X

1

and

X

2

haveanonempty interse tion.

Proof. Pi k some origin

O

in

E

. Write

X = (x

i

)

i∈L

for some index set

L

(we an let

byLemma 2.6.5from Gallier [20℄, there is a family

(µ

k

)

k∈L

(ofnite support) of s alars, notallnull,su h that

X

k∈L

µ

k

= 0

and

X

k∈L

µ

k

Ox

k

= 0.

Sin e

P

k∈L

µ

k

= 0

,the

µ

k

arenotallnull,and

(µ

k

)

k∈L

hasnitesupport,thesets

I = {i ∈ L | µ

i

> 0}

and

J = {j ∈ L | µ

j

< 0}

arenonempty,nite, andobviouslydisjoint. Let

X

1

= {x

i

∈ X | µ

i

> 0}

and

X

2

= {x

i

∈ X | µ

i

≤ 0}.

Again,sin e the

µ

k

are notallnull and

P

k∈L

µ

k

= 0

, the sets

X

1

and

X

2

arenonempty, andobviously

X

1

∩ X

2

= ∅

and

X

1

∪ X

2

= X.

Furthermore, the denition of

I

and

J

implies that

(x

i

)

i∈I

⊆ X

1

and

(x

j

)

j∈J

⊆ X

2

. It remainsto provethat

C(X

1

) ∩ C(X

2

) 6= ∅

. Thedenitionof

I

and

J

impliesthat

X

k∈L

µ

k

Ox

k

= 0

anbewrittenas

X

i∈I

µ

i

Ox

i

+

X

j∈J

µ

j

Ox

j

= 0,

thatis,as

X

i∈I

µ

i

Ox

i

=

X

j∈J

−µ

j

Ox

j

,

where

X

i∈I

µ

i

=

X

j∈J

−µ

j

= µ,

with

µ > 0

. Thus,wehave

X

i∈I

µ

i

µ

Ox

i

=

X

j∈J

−

µ

j

µ

Ox

j

,

with

X

i∈I

µ

i

µ

=

X

j∈J

−

µ

j

µ

= 1,

provingthat

P

i∈I

(µ

i

/µ)x

i

∈ C(X

1

)

and

P

j∈J

−(µ

j

/µ)x

j

∈ C(X

2

)

are identi al, andthus that

C(X

1

) ∩ C(X

2

) 6= ∅

.

Theorem2.7 Givenany anespa e

E

of dimension

m

,foreveryfamily

{K

1

, . . . , K

n

}

of

n

onvexsubsetsof

E

,if

n ≥ m + 2

andthe interse tion

T

i∈I

K

i

ofany

m + 1

of the

K

i

is nonempty (where

I ⊆ {1, . . . , n}

,

|I| = m + 1

),then

T

n

i=1

K

i

isnonempty.

Proof. Theproofisbyindu tionon

n ≥ m + 1

andusesRadon'stheoremintheindu tion step. For

n = m + 1

,theassumptionofthetheoremisthattheinterse tionofanyfamilyof

m+1

ofthe

K

i

'sisnonempty,andthetheoremholdstrivially. Next,let

L = {1, 2, . . . , n+1}

, where

n + 1 ≥ m + 2

. Bythe indu tion hypothesis,

C

i

=

T

j∈(L−{i})

K

j

is nonempty for every

i ∈ L

.

We laimthat

C

i

∩ C

j

6= ∅

forsome

i 6= j

. Ifso,as

C

i

∩ C

j

=

T

n+1

k=1

K

k

,wearedone. So, letusassumethatthe

C

i

'sarepairwisedisjoint. Then,we anpi kaset

X = {a

1

, . . . , a

n+1

}

su h that

a

i

∈ C

i

,for every

i ∈ L

. ByRadon'sTheorem, there aretwononemptydisjoint sets

X

1

, X

2

⊆ X

su h that

X = X

1

∪ X

2

and

C(X

1

) ∩ C(X

2

) 6= ∅

. However,

X

1

⊆ K

j

for every

j

with

a

j

∈ X

/

1

. Thisisbe ause

a

j

∈ K

/

j

forevery

j

,andso,weget

X

1

⊆

\

a

j

∈X

/

1

K

j

.

Symetri ally,wealsohave

X

2

⊆

\

a

j

∈X

/

2

K

j

.

Sin ethe

K

j

'sare onvexand





\

a

j

∈X

/

1

K

j



 ∩





\

a

j

∈X

/

2

K

j



 =

n+1

_\

i=1

K

i

,

itfollowsthat

C(X

1

) ∩ C(X

2

) ⊆

T

n+1

i=1

K

i

, sothat

T

n+1

i=1

K

i

is nonempty, ontradi tingthe fa tthat

C

i

∩ C

j

= ∅

forall

i 6= j

.

AmoregeneralversionofHelly'stheoremisprovedinBerger[6℄. An amusing orollary ofHelly'stheoremisthefollowingresult: Consider

n ≥ 4

parallellinesegmentsintheane plane

A

2

. Ifeverythree ofthese line segmentsmeet aline, thenallof these linesegments meeta ommonline.

We on lude this hapter with ani e appli ation of Helly's Theorem to the existen e of enterpoints. Centerpointsgeneralize thenotionofmedian tohigherdimensions. Re all that if wehave aset of

n

data points,

S = {a

1

, . . . , a

n

}

, onthe real line, a median for

S

isapoint,

x

, su hthat at least

n/2

of thepointsin

S

belong to bothintervals

[x, ∞)

and

(−∞, x]

.

Givenanyhyperplane,

H

,re allthatthe losedhalf-spa esdeterminedby

H

aredenoted

H

+

and

H

−

and that

H ⊆ H

+

and

H ⊆ H

−

. Welet

◦

H

+

= H

+

− H

and

◦

H

−

= H

−

− H

be theopen half-spa es determinedby

H

.

Denition2.5 Let

S = {a

1

, . . . , a

n

}

be aset of

n

points in

A

d

. A point,

c ∈ A

d

, is a enterpointof

S

iforeveryhyperplane,

H

,wheneverthe losedhalf-spa e

H

+

(resp.

H

−

) ontains

c

,then

H

+

(resp.

H

−

) ontainsatleast

n

d+1

pointsfrom

S

.

So, for

d = 2

,for ea h line,

D

,ifthe losedhalf-plane

D

+

(resp.

D

−

) ontains

c

, then

D

+

(resp.

D

−

) ontainsatleastathirdofthepointsfrom

S

. For

d = 3

,forea hplane,

H

, ifthe losedhalf-spa e

H

+

(resp.

H

−

) ontains

c

, then

H

+

(resp.

H

−

) ontainsatleast a fourthofthepointsfrom

S

,et .

Observethatapoint,

c ∈ A

d

,isa enterpointof

S

i

c

belongstoeveryopenhalf-spa e,

◦

H

+

(resp.

◦

H

−

) ontainingat least

dn

d+1

+ 1

pointsfrom

S

.

Indeed, if

c

is a enterpoint of

S

and

H

is anyhyperplanesu h that

◦

H

+

(resp.

◦

H

−

) ontainsatleast

dn

d+1

+ 1

pointsfrom

S

,then

◦

H

+

(resp.

◦

H

−

)must ontain

c

asotherwise, the losedhalf-spa e,

H

−

(resp.

H

+

)would ontain

c

andat most

n −

dn

d+1

− 1 =

n

d+1

− 1

pointsfrom

S

,a ontradi tion. Conversely,assumethat

c

belongstoeveryopenhalf-spa e,

◦

H

+

(resp.

◦

H

−

) ontainingat least

dn

d+1

+ 1

pointsfrom

S

. Then, foranyhyperplane,

H

, if

c ∈ H

+

(resp.

c ∈ H

−

)but

H

+

ontainsat most

n

d+1

− 1

pointsfrom

S

, then theopen half-spa e,

◦

H

−

(resp.

◦

H

+

)would ontainatleast

n −

n

d+1

+ 1 =

dn

d+1

+ 1

pointsfrom

S

but not

c

,a ontradi tion.

Wearenowreadytoprovetheexisten eof enterpoints. Theorem2.8 Every niteset,

S = {a

1

, . . . , a

n

}

,of

n

pointsin

A

d

has some enterpoint. Proof. We will use these ond hara terization of enterpoints involving open half-spa es ontainingat least

dn

d+1

+ 1

points. Considerthefamilyofsets,

C

=



conv(S ∩

H

◦

+

) | (∃H)



|S ∩

H

◦

+

| >

dn

d + 1



∪



conv(S ∩

H

◦

−

) | (∃H)



|S ∩

H

◦

−

| >

dn

d + 1



,

where

H

isahyperplane.

As

S

isnite,

C

onsistsofanitenumberof onvexsets,say

{C

1

, . . . , C

m

}

. Ifweprove that

T

m

i=1

C

i

6= ∅

wearedone,be ause

T

m

i=1

C

i

isthesetof enterpointsof

S

.

First,weprovebyindu tionon

k

(with

1 ≤ k ≤ d + 1

),thatanyinterse tionof

k

ofthe

C

i

'shasat least

(d+1−k)n

d+1

+ k

elementsfrom

S

. For

k = 1

, this holdsby denition ofthe

Next, onsider theinterse tionof

k + 1 ≤ d + 1

ofthe

C

i

's,say

C

i

1

∩ · · · ∩ C

i

k

∩ C

i

k+1

. Let

A =

S ∩ (C

i

1

∩ · · · ∩ C

i

k

∩ C

i

k+1

)

B

=

S ∩ (C

i

1

∩ · · · ∩ C

i

k

)

C

=

S ∩ C

i

k+1

.

Notethat

A = B ∩C

. Bytheindu tionhypothesis,

B

ontainsatleast

(d+1−k)n

d+1

+k

elements from

S

. As

C

ontainsat least

dn

d+1

+ 1

pointsfrom

S

, andas

|B ∪ C| = |B| + |C| − |B ∩ C| = |B| + |C| − |A|

and

|B ∪ C| ≤ n

,weget

n ≥ |B| + |C| − |A|

,thatis,

|A| ≥ |B| + |C| − n.

Itfollowsthat

|A| ≥

(d + 1 − k)n

d + 1

+ k +

dn

d + 1

+ 1 − n

thatis,

|A| ≥

(d + 1 − k)n + dn − (d + 1)n

d + 1

+ k + 1 =

(d + 1 − (k + 1))n

d + 1

+ k + 1,

establishingtheindu tionhypothesis.

Now,if

m ≤ d + 1

,theabove laimfor

k = m

showsthat

T

m

i=1

C

i

6= ∅

andwearedone. If

m ≥ d + 2

,theabove laimfor

k = d + 1

showsthatanyinterse tionof

d + 1

ofthe

C

i

's is nonempty. Consequently, the onditionsforapplying Helly's Theorem are satisedand therefore,

m

\

i=1

C

i

6= ∅.

However,

T

m

i=1

C

i

isthesetof enterpointsof

S

andwearedone.

Remark: Theaboveproofa tuallyshowsthatthesetof enterpointsof

S

isa onvexset. Infa t,itisanite interse tionof onvexhulls ofnitelymanypoints,soitisthe onvex hullofnitely manypoints,in otherwords,apolytope.

Jadhavand Mukhopadhyayhavegivenalinear-timealgorithmfor omputinga enter-pointof anite setof pointsin theplane. For

d ≥ 3

,it appearsthat thebestthat anbe done (using linearprogramming) is

O(n

d

₎

. However, there are good approximation algo-rithms(Clarkson,Eppstein,Miller,SturtivantandTeng)andin

E

3

thereisanearquadrati algorithm(Agarwal,SharirandWelzl).

Chapter 3

Separation and Supporting

Hyperplanes

3.1 Separation Theorems and Farkas Lemma

It seemsintuitively ratherobviousthat if

A

and

B

are twononempty disjoint onvexsets in

A

2

,thenthereisaline,

H

,separatingthem,inthesensethat

A

and

B

belongtothetwo (disjoint)openhalfplanesdeterminedby

H

. However,thisisnotalwaystrue! Forexample, this failsif both

A

and

B

are losed and unbounded (ndanexample). Nevertheless,the resultis trueifboth

A

and

B

are open, orifthe notionofseparationis weakenedalittle bit. The keyresult, from whi h most separationresults follow, is ageometri version of theHahn-Bana h theorem. Inthesequel, werestri t ourattention toreal ane spa esof nitedimension. Then,if

X

isananespa eofdimension

d

,thereisananebije tion

f

between

X

and

A

d

. Now,

A

d

isatopologi al spa e,under theusual topologyon

R

d

(in fa t,

A

d

isametri spa e). Re all that if

a = (a

1

, . . . , a

d

)

and

b = (b

1

, . . . , b

d

)

are anytwopointsin

A

d

, their Eu lidean distan e,

d(a, b)

,isgivenby

d(a, b) =

p

(b

1

− a

1

)

2

+ · · · + (b

d

− a

d

)

2

,

whi h is also the norm,

kabk

, of the ve tor

ab

and that for any

ǫ > 0

, the open ball of enter

a

andradius

ǫ

,

B(a, ǫ)

,is givenby

B(a, ǫ) = {b ∈ A

d

| d(a, b) < ǫ}.

A subset

U ⊆ A

d

is open (in the norm topology) if either

U

is empty orfor everypoint,

a ∈ U

,there issome(small)openball,

B(a, ǫ)

, ontainedin

U

. A subset

C ⊆ A

d

is losed i

A

d

_{− C}

isopen. Forexample,the losedballs,

B(a, ǫ)

,where

are losed. A subset

W ⊆ A

d

isbounded ithereis someball(openor losed),

B

,so that

W ⊆ B

. Asubset

W ⊆ A

d

is ompa t ieveryfamily,

{U

i

}

i∈I

,thatisanopen overof

W

(whi hmeansthat

W =

S

i∈I

(W ∩ U

i

)

,withea h

U

i

anopenset)possessesanitesub over (whi h meansthat there isanite subset,

F ⊆ I

, sothat

W =

S

i∈F

(W ∩ U

i

)

). In

A

d

, it anbeshownthat asubset

W

is ompa ti

W

is losedand bounded. Given afun tion,

f : A

m

_{→ A}

n

, wesaythat

f

is ontinuous if

f

−1

_{(V )}

is openin

A

m

whenever

V

isopen in

A

n

. If

f : A

m

_{→ A}

n

is a ontinuous fun tion, although it is generally false that

f (U )

is openif

U ⊆ A

m

isopen,itiseasily he kedthat

f (K)

is ompa tif

K ⊆ A

m

is ompa t. An anespa e

X

ofdimension

d

be omesatopologi alspa eifwegiveitthetopology forwhi htheopen subsetsareofthe form

f

−1

_{(U )}

,where

U

isanyopensubsetof

A

d

and

f : X → A

d

isananebije tion.

Givenanysubset,

A

, ofatopologi al spa e,

X

, thesmallest losed set ontaining

A

is denotedby

A

,andis alledthe losure oradheren eof

A

. Asubset,

A

,of

X

,isdensein

X

if

A = X

. Thelargestopenset ontainedin

A

isdenotedby

◦

A

,andis alledtheinteriorof

A

. Theset,

Fr A = A ∩ X − A

, is alled theboundary (or frontier)of

A

. We alsodenote theboundaryof

A

by

∂A

.

Inorder to provethe Hahn-Bana htheorem, wewillneed twolemmas. Givenanytwo distin tpoints

x, y ∈ X

,welet

]x, y[ = {(1 − λ)x + λy ∈ X | 0 < λ < 1}.

Ourrstlemma(Lemma3.1)isintuitivelyquiteobvioussothereadermightbepuzzledby thelengthofitsproof. However,afterproposingseveral wrongproofs,werealized thatits proofismoresubtlethanitmightappear. Theproofbelowisdue toValentine[41℄. Seeif you anndashorter(and orre t)proof!

Lemma3.1 Let

S

be a nonempty onvex set and let

x ∈

◦

S

and

y ∈ S

. Then, we have

]x, y[ ⊆

S

◦

.

Proof. Let

z ∈ ]x, y[

, that is,

z = (1 − λ)x + λy

, with

0 < λ < 1

. Sin e

x ∈

◦

S

, we an nd some open subset,

U

, ontained in

S

so that

x ∈ U

. It is easy to he k that the entralmagni ationof enter

z

,

H

z,

λ−1

_λ

,maps

x

to

y

. Then,

H

z,

λ−1

_λ

(U )

isanopensubset ontaining

y

andas

y ∈ S

,wehave

H

z,

λ−1

λ

(U ) ∩ S 6= ∅

. Let

v ∈ H

z,

λ−1

λ

(U ) ∩ S

beapoint of

S

in thisinterse tion. Now,thereisauniquepoint,

u ∈ U ⊆ S

,su hthat

H

z,

λ−1

_λ

(u) = v

and,as

S

is onvex,wededu ethat

z = (1 − λ)u + λv ∈ S

. Sin e

U

isopen,theset

(1 − λ)U + λv = {(1 − λ)w + λv | w ∈ U } ⊆ S

isalsoopenand

z ∈ (1 − λ)U + λv

, whi hshowsthat

z ∈

◦

Corollary3.2 If

S

is onvex, then

◦

S

is also onvex, andwe have

◦

S =

◦

S

. Furthermore, if

◦

S 6= ∅

,then

S =

◦

S

.



Beware thatif

S

isa losedset,then the onvexhull,

conv(S)

, of

S

is notne essarily losed! (Finda ounter-example.) However,it anbeshownthatif

S

is ompa t,then

conv(S)

isalso ompa tandthus, losed.

Thereisasimple riteriontotestwhether a onvexsethasanemptyinterior,basedon thenotionofdimensionofa onvexset.

Denition3.1 The dimension ofanonempty onvexsubset,

S

, of

X

,denoted by

dim S

, isthedimensionofthesmallestanesubset,

hSi

, ontaining

S

.

Proposition 3.3 A nonempty onvexset

S

has anonempty interiori

dim S = dim X

. Proof. Let

d = dim X

. First,assumethat

◦

S 6= ∅

. Then,

S

ontainssomeopenballof enter

a

0

,andin it,we anndaframe

(a

0

, a

1

, . . . , a

d

)

for

X

. Thus,

dim S = dim X

. Conversely, let

(a

0

, a

1

, . . . , a

d

)

beaframeof

X

,with

a

i

∈ S

,for

i = 0, . . . , d

. Then,wehave

a

0

+ · · · + a

d

d + 1

∈

◦

S,

and

◦

S

isnonempty.



Proposition3.3isfalseininnitedimension.

Weleavethefollowingpropertyasanexer ise: Proposition 3.4 If

S

is onvex, then

S

isalso onvex.

One an also easily prove that onvexity is preserved under dire t image and inverse imagebyananemap.

Thenextlemma, whi h seemsintuitivelyobvious,isthe oreoftheproof ofthe Hahn-Bana htheorem. Thisisthe asewheretheanespa ehasdimensiontwo. First,weneed todenewhat isa onvex one.

Denition3.2 A onvexset,

C

, isa onvex one with vertex

x

if

C

is invariantunderall entralmagni ations,

H

x,λ

,of enter

x

andratio

λ

, with

λ > 0

(i.e.,

H

x,λ

(C) = C

).

Givena onvexset,

S

,andapoint,

x /

∈ S

,we andene

cone

x

(S) =

[

λ>0

H

x,λ

(S).

B

O

C

x

L

Figure3.1: Hahn-Bana hTheoremintheplane(Lemma3.5) Lemma3.5 Let

B

be anonempty open and onvex subset of

A

2

,and let

O

be apoint of

A

2

sothat

O /

∈ B

. Then,there issomeline,

L

,through

O

,sothat

L ∩ B = ∅

.

Proof. Dene the onvex one

C = cone

O

(B)

. As

B

isopen,it iseasyto he kthatea h

H

O,λ

(B)

is open and sin e

C

is theunionof the

H

O,λ

(B)

(for

λ > 0

),whi h areopen,

C

itselfisopen. Also,

O /

∈ C

. We laimthat aleast onepoint,

x

,oftheboundary,

∂C

,of

C

, is distin tfrom

O

. Otherwise,

∂C = {O}

and we laimthat

C = A

2

_{− {O}}

,whi h is not onvex,a ontradi tion. Indeed,as

C

is onvexitis onne ted,

A

2

_{− {O}}

itselfis onne ted and

C ⊆ A

2

_{− {O}}

. If

C 6= A

2

_{− {O}}

, pi ksome point

a 6= O

in

A

2

_{− C}

and somepoint

c ∈ C

. Now,abasi propertyof onne tivityassertsthatevery ontinuouspathfrom

a

(in theexteriorof

C

)to

c

(intheinteriorof

C

)mustinterse ttheboundaryof

C

,namely,

{O}

. However, there are plenty of paths from

a

to

c

that avoid

O

, a ontradi tion. Therefore,

C = A

2

_{− {O}}

.

Sin e

C

isopenand

x ∈ ∂C

,wehave

x /

∈ C

. Furthermore,we laimthat

y = 2O − x

(the symmetri of

x

w.r.t.

O

)doesnotbelongto

C

either. Otherwise,wewouldhave

y ∈

◦

C = C

and

x ∈ C

, and byLemma 3.1,wewouldget

O ∈ C

, a ontradi tion. Therefore,the line through

O

and

x

misses

C

entirely(sin e

C

isa one),andthus,

B ⊆ C

.

Finally,we ometotheHahn-Bana htheorem.

Theorem3.6 (Hahn-Bana h Theorem, geometri form) Let

X

be a (nite-dimensional) anespa e,

A

beanonempty open and onvexsubset of

X

and

L

be anane subspa e of

X

sothat

A ∩ L = ∅

. Then,thereissomehyperplane,

H

, ontaining

L

,thatisdisjointfrom

A

.

Proof. The asewhere

dim X = 1

is trivial. Thus, wemayassumethat

dim X ≥ 2

. We redu etheprooftothe asewhere

dim X = 2

. Let

V

beananesubspa eof

X

ofmaximal

A

L

H

Figure3.2: Hahn-Bana hTheorem, geometri form(Theorem3.6)

dimension ontaining

L

andsothat

V ∩ A = ∅

. Pi kanorigin

O ∈ L

in

X

,and onsiderthe ve torspa e

X

O

. Wewould liketo provethat

V

isahyperplane,i.e.,

dim V = dim X − 1

. We pro eed by ontradi tion. Thus, assume that

dim V ≤ dim X − 2

. In this ase, the quotient spa e

X/V

has dimensionat least

2

. We also know that

X/V

is isomorphi to theorthogonal omplement,

V

⊥

,of

V

sowemayidentify

X/V

and

V

⊥

. The(orthogonal) proje tionmap,

π : X → V

⊥

, islinear, ontinuous,andwe anshowthat

π

mapstheopen subset

A

toanopensubset

π(A)

,whi hisalso onvex(onewaytoprovethat

π(A)

isopenis toobservethatforanypoint,

a ∈ A

,asmallopenballof enter

a

ontainedin

A

isproje ted by

π

toanopenball ontainedin

π(A)

and as

π

issurje tive,

π(A)

isopen). Furthermore,

0 /

∈ π(A)

. Sin e

V

⊥

hasdimensionatleast

2

,thereissomeplane

P

(asubspa eofdimension

2

)interse ting

π(A)

,andthus,weobtainanonemptyopenand onvexsubset

B = π(A) ∩ P

in theplane

P ∼

= A

2

. So, we an applyLemma 3.5 to

B

andthe point

O = 0

in

P ∼

= A

2

tondaline,

l

,(in

P

)through

O

with

l ∩ B = ∅

. Butthen,

l ∩ π(A) = ∅

and

W = π

−1

_(l)

is anane subspa esu h that

W ∩ A = ∅

and

W

properly ontains

V

, ontradi tingthe maximalityof

V

.

Remark: Thegeometri formoftheHahn-Bana htheoremalsoholdswhenthedimension of

X

isinnitebutaslightlymoresophisti atedproofisrequired. A tually,allthatisneeded is to provethat a maximal ane subspa e ontaining

L

and disjoint from

A

exists. This anbedone usingZorn'slemma. Forother proofs, seeBourbaki [9℄, Chapter 2,Valentine [41℄,Chapter2,Barvinok[3℄,Chapter 2,orLax[26℄, Chapter3.



Theorem3.6isfalseifweomittheassumptionthat

A

isopen. Fora ounter-example, let

A ⊆ A

2

A

L

H

Figure3.3: Hahn-Bana hTheorem,se ondversion(Theorem3.7)

x

-axis and let

L

be the point

(2, 0)

onthe boundary of

A

. It is also false if

A

is losed! (Finda ounter-example).

Theorem3.6hasmanyimportant orollaries. Forexample,wewilleventuallyprovethat for any two nonempty disjoint onvexsets,

A

and

B

, there is ahyperplaneseparating

A

and

B

, butthis willtakesomework(re allthedenitionof aseparatinghyperplanegiven inDenition 2.1). WebeginwiththefollowingversionoftheHahn-Bana htheorem: Theorem3.7 (Hahn-Bana h,se ondversion)Let

X

bea(nite-dimensional)anespa e,

A

beanonempty onvexsubsetof

X

with nonemptyinterior and

L

beananesubspa eof

X

so that

A ∩ L = ∅

. Then, there is some hyperplane,

H

, ontaining

L

andseparating

L

and

A

.

Proof. Sin e

A

is onvex,byCorollary3.2,

◦

A

isalso onvex.Byhypothesis,

◦

A

isnonempty. So,we anapplyTheorem3.6tothenonemptyopenand onvex

◦

A

andtotheanesubspa e

L

. Wegetahyperplane

H

ontaining

L

su hthat

◦

A ∩ H = ∅

. However,

A ⊆ A =

◦

A

and

◦

A

is ontainedin the losedhalfspa e(

H

+

or

H

−

) ontaining

◦

A

,so

H

separates

A

and

L

. Corollary3.8 Given an ane spa e,

X

, let

A

and

B

be two nonempty disjoint onvex subsets and assume that

A

has nonempty interior (

◦

A 6= ∅

). Then, there is a hyperplane separating

A

and

B

.

A

B

H

Figure3.4: SeparationTheorem,version1(Corollary3.8)

Proof. Pi ksomeorigin

O

and onsidertheve torspa e

X

O

. Dene

C = A − B

(aspe ial aseoftheMinkowskisum)asfollows:

A − B = {a − b | a ∈ A, b ∈ B} =

[

b∈B

(A − b).

Itiseasilyveriedthat

C = A−B

is onvexandhasnonemptyinterior(asaunionofsubsets havinganonemptyinterior). Furthermore

O /

∈ C

,sin e

A∩B = ∅

.

1

(Notethatthedenition depends onthe hoi e of

O

, but thishasnoee t ontheproof.) Sin e

◦

C

isnonempty, we anapply Theorem3.7 to

C

and to theanesubspa e

{O}

andwegeta hyperplane,

H

, separating

C

and

{O}

. Let

f

beanylinearformdeningthehyperplane

H

. Wemayassume that

f (a − b) ≤ 0

, for all

a ∈ A

and all

b ∈ B

, i.e.,

f (a) ≤ f (b)

. Consequently, if we let

α = sup{f (a) | a ∈ A}

(whi hmakessense,sin etheset

{f (a) | a ∈ A}

isbounded),wehave

f (a) ≤ α

for all

a ∈ A

and

f (b) ≥ α

forall

b ∈ B

, whi h showsthat theanehyperplane denedby

f − α

separates

A

and

B

.

Remark: Theorem3.7andCorollary3.8alsoholdintheinnitedimensional ase,seeLax [26℄,Chapter3,orBarvinok,Chapter3.

1

Readerswhopreferapurelyaneargumentmaydene

C = A − B

astheanesubset

A − B = {O + a − b | a ∈ A, b ∈ B}.

Again,

O /

∈ C

and

C

is onvex. Byadjusting

O

we an pi k the aneform,

f

, dening a separating hyperplane,

H

,of

C

and

{O}

,sothat

f (O + a − b) ≤ f (O)

,forall

a ∈ A

andall

b ∈ B

,i.e.,

f (a) ≤ f (b)

.

Sin eahyperplane,

H

,separating

A

and

B

asin Corollary3.8 istheboundaryof ea h ofthetwohalfspa esthatitdetermines,wealsoobtainthefollowing orollary:

Corollary3.9 Given anane spa e,

X

,let

A

and

B

be twononempty disjoint open and onvexsubsets. Then, thereisahyperplane stri tlyseparating

A

and

B

.



Bewarethat Corollary3.9 fails for losed onvexsets. However,Corollary3.9holdsif wealsoassumethat

A

(or

B

)is ompa t.

Weneed to reviewthenotionof distan e from apointto asubset. Let

X

be ametri spa ewith distan efun tion,

d

. Given anypoint,

a ∈ X

, andanynonemptysubset,

B

, of

X

,welet

d(a, B) = inf

b∈B

d(a, b)

(where

inf

isthenotationforleastupperbound).

Now,if

X

isananespa eofdimension

d

,it anbegivenametri stru turebygiving the orresponding ve tor spa e a metri stru ture, for instan e, the metri indu ed by a Eu lideanstru ture. We havethe followingimportantproperty: Foranynonempty losed subset,

S ⊆ X

(not ne essarily onvex), and any point,

a ∈ X

, there is somepoint

s ∈ S

a hievingthedistan efrom

a

to

S

, i.e.,so that

d(a, S) = d(a, s).

The proof uses the fa t that the distan e fun tion is ontinuous and that a ontinuous fun tionattainsitsminimumona ompa tset,andisleft asanexer ise.

Corollary3.10 Given an ane spa e,

X

, let

A

and

B

be two nonempty disjoint losed and onvex subsets,with

A

ompa t. Then, there isahyperplane stri tly separating

A

and

B

.

Proof sket h. First,wepi kanorigin

O

andwegive

X

O

∼

= A

n

aEu lideanstru ture. Let

d

denotetheasso iateddistan e. Givenanysubsets

A

of

X

,let

A + B(O, ǫ) = {x ∈ X | d(x, A) < ǫ},

where

B(a, ǫ)

denotestheopenball,

B(a, ǫ) = {x ∈ X | d(a, x) < ǫ}

,of enter

a

andradius

ǫ > 0

. Notethat

A + B(O, ǫ) =

[

a∈A

B(a, ǫ),

whi hshowsthat

A + B(O, ǫ)

isopen;furthermoreitiseasytoseethatif

A

is onvex,then

A + B(O, ǫ)

isalso onvex. Now,thefun tion

a 7→ d(a, B)

(where

a ∈ A

)is ontinuousand sin e

A

is ompa t, ita hievesitsminimum,

d(A, B) = min

a∈A

d(a, B)

, at somepoint,

a

,

of

A

. Say,

d(A, B) = δ

. Sin e

B

is losed, thereissome

b ∈ B

so that

d(A, B) = d(a, B) =

d(a, b)

andsin e

A ∩ B = ∅

,wemusthave

δ > 0

. Thus,ifwepi k

ǫ < δ/2

,weseethat

(A + B(O, ǫ)) ∩ (B + B(O, ǫ)) = ∅.

Now,

A+B(O, ǫ)

and

B +B(O, ǫ)

areopen, onvexanddisjointandwe on ludebyapplying Corollary3.9.

A  ute appli ation of Corollary 3.10is one of the many versions of Farkas Lemma (1893-1894, 1902), a basi result in the theory of linear programming. For any ve tor,

x = (x

1

, . . . , x

n

) ∈ R

n

,andanyreal,

α ∈ R

, write

x ≥ α

i

x

i

≥ α

,for

i = 1, . . . , n

.

Lemma3.11 (Farkas Lemma,VersionI)Given any

d × n

realmatrix,

A

,andany ve tor,

z ∈ R

d

,exa tly oneof the following alternativeso urs:

(a) The linear system,

Ax = z

, has a solution,

x = (x

1

, . . . , x

n

)

, su h that

x ≥ 0

and

x

1

+ · · · + x

n

= 1

,or (b) Thereissome

c ∈ R

d

andsome

α ∈ R

su hthat

c

⊤

_{z < α}

and

c

⊤

_{A ≥ α}

. Proof. Let

A

1

, . . . , A

n

∈ R

d

be the

n

points orresponding to the olumns of

A

. Then, either

z ∈ conv({A

1

, . . . , A

n

})

or

z /

∈ conv({A

1

, . . . , A

n

})

. In the rst ase, we have a onvex ombination

z = x

1

A

1

+ · · · + x

n

A

n

where

x

i

≥ 0

and

x

1

+ · · · + x

n

= 1

,so

x = (x

1

, . . . , x

n

)

isasolutionsatisfying(a).

Inthese ond ase,byCorollary3.10,thereisahyperplane,

H

,stri tlyseparating

{z}

and

conv({A

1

, . . . , A

n

})

,whi hisobviously losed. Infa t,observethat

z /

∈ conv({A

1

, . . . , A

n

})

ithereis ahyperplane,

H

,su hthat

z ∈

◦

H

−

and

A

i

∈ H

+

,for

i = 1, . . . , n

. Astheane hyperplane,

H

,isthezerolo usofanequation oftheform

c

1

y

1

+ · · · + c

d

y

d

= α,

either

c

⊤

_{z < α}

and

c

⊤

_A

i

≥ α

for

i = 1, . . . , n

, thatis,

c

⊤

_{A ≥ α}

,or

c

⊤

_{z > α}

and

c

⊤

_{A ≤ α}

. Inthese ond ase,

(−c)

⊤

_{z < −α}

and

(−c)

⊤

_{A ≥ −α}

,so(b)issatisedbyeither

c

and

α

or by

−c

and

−α

.

Remark: If we relax the requirements on solutions of

Ax = z

and only require

x ≥ 0

(

x

1

+ · · · + x

n

= 1

isno longer required)then, in ondition(b), we antake

α = 0

. This isanotherversionofFarkasLemma. Inthis ase, insteadof onsideringthe onvexhullof

{A

1

, . . . , A

n

}

weare onsideringthe onvex one,

cone(A

1

, . . . , A

n

) = {λA

1

+ · · · + λ

n

A

n

| λ

i

≥ 0, 1 ≤ i ≤ n},

thatis,wearedroppingthe ondition

λ

1

+ · · · + λ

n

= 1

. ForthisversionofFarkasLemma weneedthefollowingseparationlemma: