Do Minkowski averages get progressively more convex?

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Do Minkowski averages get progressively more convex?

Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, Artem Zvavitch

To cite this version:

Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, Artem Zvavitch. Do Minkowski averages

get progressively more convex?. Comptes Rendus. Mathématique, Académie des sciences (Paris),

2016, 354, pp.185 - 189. �10.1016/j.crma.2015.12.005�. �hal-01590072�

Contents lists available atScienceDirect

C. R. Acad. Sci. Paris, Ser. I

www.sciencedirect.com

Functional analysis/Geometry

Do Minkowski averages get progressively more convex?

Les moyennes de Minkowski deviennent-elles progressivement plus convexes ?

Matthieu Fradelizi^a,1,Mokshay Madiman^b,2,Arnaud Marsiglietti^c,3, Artem Zvavitch^d,4

aLaboratoired’analyseetdemathématiquesappliquées,UMR8050,UniversitéParis-EstMarne-la-Vallée,5,bdDescartes, Champs-sur-Marne,77454Marne-la-Valléecedex2,France

bUniversityofDelaware,DepartmentofMathematicalSciences,501EwingHall,Newark,DE19716,USA

cInstituteforMathematicsanditsApplications,UniversityofMinnesota,207ChurchStreetSE,434LindHall,Minneapolis,MN 55455,USA dDepartmentofMathematicalSciences,KentStateUniversity,Kent,OH44242,USA

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received6November2015

Acceptedafterrevision11December2015 Availableonline27January2016 PresentedbyGillesPisier

Letusdefine,foracompactset A⊂^{Rn,theMinkowskiaveragesofA:}

A(k)=

a1+ · · · +a_k

k :a1, . . . ,ak∈A

=¹ k

A+ · · · +A

ktimes

.

We study the monotonicity of the convergence of A(k) towards the convex hull of A, when considering the Hausdorff distance, the volume deficit and a non-convexity index of Schneider as measures ofconvergence. For the volume deficit, we show that monotonicityfailsingeneral,thusdisprovingaconjectureofBobkov,MadimanandWang.

ForSchneider’snon-convexityindex,weprovethatastrongformofmonotonicityholds, andfortheHausdorffdistance,weestablishthatthesequenceiseventuallynonincreasing.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Pourtoutensemblecompact A⊂^{Rn,définissonssesmoyennesdeMinkowskipar} A(k)=

a1+ · · · +ak

k :a1, . . . ,ak∈A

=¹ k

A+ · · · +A

kfois

.

Nous étudions lamonotonie dela convergencede A(k) versl’enveloppeconvexe de A, mesuréeparladistancedeHausdorff,ledéficitvolumiqueetparl’indicedenon-convexité

E-mailaddresses:matthieu.fradelizi@u-pem.fr(M. Fradelizi),madiman@udel.edu(M. Madiman),arnaud.marsiglietti@ima.umn.edu(A. Marsiglietti), zvavitch@math.kent.edu(A. Zvavitch).

1 SupportedinpartbytheAgenceNationaledelaRecherche,projectGeMeCoD(ANR2011BS0100701).

2 SupportedinpartbytheU.S.NationalScienceFoundationthroughthegrantsDMS-1409504(CAREER)andCCF-1346564.

3 SupportedinpartbytheInstituteforMathematicsanditsApplicationswithfundsprovidedbytheNationalScienceFoundation.

4 SupportedinpartbytheU.S.NationalScienceFoundationGrantDMS-1101636andbytheSimonsFoundation.

http://dx.doi.org/10.1016/j.crma.2015.12.005

1631-073X/©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

186 M. Fradelizi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 185–189

deSchneider.Pourledéficitvolumique,nousdémontronsquelapropriétédemonotonie n’estpassatisfaiteengénéral,réfutantainsiuneconjecturedeBobkov,MadimanetWang.

Pourl’index de non-convexité de Schneider,nous montrons unepropriété renforcée de monotonie, tandis que,pour ladistance de Hausdorff, nousétablissons quela suiteest décroissanteàpartird’uncertainrang.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Versionfrançaiseabrégée

L’objectifdecettenoteestd’annonceretdedémontrerunepartiedesrésultatsobtenusdans[3]quiportentsurl’étude delamonotoniedelasuite(A(k))k≥¹ définieen(1),mesuréeàtraversdifférentesmesuresdenon-convexité.Intuitivement, les ensembles A(k)deviennentde plusenplus convexesaufuretà mesurequek croît.Cette intuitionestpréciséedans [7,2]oùilestdémontréquelasuite(A(k))convergeverssonenveloppeconvexeendistancedeHausdorffdH.

L’origine de notre étude provient d’une conjecture de Bobkov, Madiman et Wang [1], qui affirme que la suite ((A(k)))k≥¹ estdécroissante,où

(A):=^Voln(conv(A)\^A)=^Voln(conv(A))−^Voln(A)

désigneledéficitvolumiqued’unensemblecompactdeRⁿ.Ici,Voln représentelamesuredeLebesguedansRⁿ etconv(A) désigne l’enveloppe convexede A. Nousréfutons cetteconjecture enexhibant un contre-exempleexplicite endimension supérieureouégaleà12.Lecontre-exempleestlaréuniondedeuxensemblesconvexesinclusdansdessous-espacesdedi- mension(presque)moitiédel’espaceambiant(voirFig. 1).Nousdémontronsaussilavaliditédelaconjectureendimension 1enadaptantunedémonstrationde[4]surlecardinaldesommesd’entiers ;celaaaussiétéobservéindépendammentpar F. Barthe.Laconjectureresteouverteendimensionn,pour1<n<12.

Demanière analogueàlaconjecturede Bobkov–Madiman–Wang,nousétudionslamonotoniede lasuite(c(A(k)))_k≥1, oùc estl’indexdenon-convexitédeSchneider[6]définipar

c(A):=^inf{λ≥⁰:^A+λconv(A)est convexe}.

Contrairementaudéficitvolumique,lasuite(c(A(k)))eststrictementdécroissante,àmoinsque A(k)soitdéjàconvexe.Plus précisémentnousmontronsquepourtoutensemblecompact A deRⁿettoutk∈N^∗

c(A(k+¹))≤ ^k

k+^1c(A(k)) .

En outre, nous étudions dans[3] la monotoniede A(k), mesurée par d’autres mesures de non-convexité. Ainsi, nous montronsquesil’onpose

d(A)=dH(A,conv(A))=inf{r>0:conv(A)⊂A+r Bⁿ₂},

où Bⁿ₂estlabouleeuclidiennecentréeen0 derayon1,alorspourtoutcompact A deRⁿ etpourk≥^c(A),

d(A(k+1))≤ ^k

k+1d(A(k)) .

1. Introduction

Thisnote announcesandprovessome oftheresultsobtainedin[3].Letusdenoteforacompactset A⊂^{Rn andfora} positiveintegerk,

A(k)=

a1+ · · · +^ak

k :^a1, . . . ,a_k∈^A

=¹ k

A+ · · · +^A

ktimes

. (1)

Denotingbyconv(A)theconvexhullofA,andby d(A):=inf{r>0:conv(A)⊂A+r Bⁿ₂}

the Hausdorff distancebetweenaset A anditsconvexhull, it isa classicalfact (provedindependentlyby [7,2] in1969, and oftencalledthe Shapley–Folkmann–Starrtheorem) that A(k) convergesin Hausdorffdistance to conv(A) ask→ ∞^. Furthermore[7,2] alsodetermined therateofconvergence:itturnsout thatd(A(k))=^O(1/k) foranycompactset A.For setsofnonemptyinterior,thisconvergenceofMinkowskiaveragestotheconvexhullcanalsobeexpressedintermsofthe volumedeficit(A)ofacompactset A inRⁿ,whichisdefinedas:

(A):=^Voln(conv(A)\^A)=^Voln(conv(A))−^Voln(A),

whereVoln denotesLebesgue measureinRⁿ.It was shownby[2]that if A is compactwithnonemptyinterior,thenthe volumedeficitofA(k)alsoconvergesto0;moreprecisely,(A(k))=^O(1/k)foranycompactsetAwithnonemptyinterior.

OuroriginalmotivationcamefromaconjecturemadebyBobkov,MadimanandWang[1]:

Conjecture1.(See[1].) LetA beacompactsetinRⁿforsomen∈N^,andletA(k)bedefinedasin(1).Thenthesequence(A(k))is non-increasingink,orequivalently,{^Voln(A(k))}k≥¹isnon-decreasing.

WeshowthatConjecture 1failstoholdingeneral,evenformoderatelyhighdimension.

Theorem2.Conjecture 1isfalseinRⁿforn≥^{12,andtruefor}R^1.

Noticethat Conjecture 1remains openfor1<n<12.Inparticular,theargumentspresentedinthisnotedonotseem towork.InanalogywithConjecture 1,wealsoconsiderwhetheronecanhavemonotonicityof{^c(A(k))}k≥¹,wherec isa non-convexityindexdefinedbySchneider[6]asfollows:

c(A):=^inf{λ≥⁰:^A+λconv(A)is convex}.

AnicepropertyofSchneider’sindexisthatitisaffine-invariant,i.e.,c(T A+^x)=^c(A)foranynonsingularlinearmap T on Rⁿandanyx∈^Rn.

Contrarytothevolumedeficit,weprovethatSchneider’snon-convexityindexc satisfiesa strongkindofmonotonicity inanydimension.

Theorem3.LetA beacompactsetinRⁿandk∈N^∗.Then c(A(k+¹))≤ ^k

k+^1c(A(k)) .

Finally,we alsoprove thateventually,fork≥^c(A), theHausdorff distancebetween A(k) andconv(A) isalsostrongly decreasing.

Theorem4.LetA beacompactsetinRⁿandk≥^c(A)beaninteger.Then d(A(k+¹))≤ ^k

k+^1d(A(k)) .

Moreover,Schneiderproved in[6]that c(A)≤^{nforevery compactsubset A ofRn.Itfollowsthat theeventual mono-} tonicityofthesequenced(A(k))holdstruefork≥^n.

Itisnaturaltoaskwhattherelationshipisingeneralbetweenconvergenceofc,anddto0,forarbitrarysequences (Ck) ofcompact sets. In fact, none of thesethree notions of approach to convexity are comparable witheach other in general. Toseewhy,observethat whilec isscaling-invariant, neithernord are;so itiseasy toconstructexamples of sequences(C_k)suchthatc(C_k)→^{0 but} (C_k)andd(C_k)remainboundedawayfrom0.Thesameargumentenablesusto constructexamples ofsequences (Ck)such that c(Ck) remain boundedaway from0,whereas (Ck) andd(Ck) converge to0.Furthermore,(Ck)remainsboundedawayfrom0foranysequenceCk offinitesets,whereas c(Ck)andd(Ck)could convergeto0ifthefinitesets formafinerandfinergridfillingouta convexset.Anexamplewhere(C_k)→^{0 butboth} c(C_k) andd(C_k)are boundedaway from0isgivenby takinga 3-pointset with2ofthepoints getting arbitrarilycloser butstayingawayfromthethird.Onecanobtainfurtherrelationships betweenthesemeasuresofnon-convexity iffurther conditionsareimposedonthesequenceC_k;detailsmaybefoundin[3].

Therestofthisnote isdevotedtotheexaminationofwhether A(k)becomesprogressivelymoreconvexaskincreases, whenmeasuredthroughthefunctionals,dandc.Theconcludingsectioncontainssomeadditionaldiscussion.

2. Thebehaviorofvolumedeficit

WeproveTheorem 2inthissection.WestartbyconstructingacounterexampletotheconjectureinRⁿ,forn≥^12.LetF bea p-dimensionalsubspaceofRⁿ,wherep∈ {¹,. . . ,n−¹}^{.LetusconsiderA}=^I1∪^I2,whereI1⊂^{F andI}2⊂^F⊥,where F^⊥denotestheorthogonalcomplementofF.Onehas(seeFig. 1):

A+^A=^2I1∪(I₁×^I2)∪^2I2,

A+^A+^A=^3I1∪(2I1×^I2)∪(I1×^2I2)∪^3I2.

Noticethat

188 M. Fradelizi et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 185–189

Fig. 1.A counterexample inR¹².

Vol_n(A+^A)=^Volp(I₁)Vol_n−p(I₂),

Vol_n(A+^A+^A)=^Volp(I₁)Vol_n−p(I₂)(2^p+^2n−p−¹).

Thus,Voln(A(3))≥^Voln(A(2))ifandonlyif 2^p+^2n−p−¹≥

3 2

_n

. (2)

Noticethatinequality(2)doesnotholdwhenn≥^{12 and p}= ⁿ₂^.

ForR¹,theconjecture maybe provedbyadapting aproof of[4]oncardinalityofintegersumsets;thiswas alsoinde- pendentlyobservedbyF. Barthe.Letk≥^1.SetS=^A1+ · · · +^Akandfori∈ [^k]^,letai=^minAi,bi=^maxAi,

Si=

j∈[^k]\{ⁱ} Aj,

si=

j<iaj+

j>ibj,S⁻_i = {^x∈^Si;^x≤^si}^andS+_i = {^x∈^Si;^x>si}^.Foralli∈ [^k−¹]^,onehas S⊃(ai+S⁻_i )∪(bi+¹+S⁺_i+1).

Sinceai+^si=

j≤ⁱaj+

j>ibj=^bi+¹+^si+¹,theaboveunionisadisjointunion.Thusfori∈ [^k−¹] Vol1(S)≥Vol1(ai+S⁻_i )+Vol1(bi+¹+S⁺_i+1)=Vol1(S⁻_i )+Vol1(S⁺_i+1).

Noticethat S⁻₁ =^S1 andS_k⁺=^Sk\ {^sk}^{,thusaddingtheabovek}−1 inequalities,weobtain

(k−¹)Vol₁(S)≥

k−1

i=¹

Vol₁(S⁻_i )+^Vol1(S⁺_i+1)

=^Vol1(S⁻₁)+^Vol1(S_k⁺)+

k−1

i=²

Vol₁(S_i)

= k

i=1

Vol₁(S_i).

Now takingallthesets A_i=^{A,anddividingthroughbyk}(k−¹),weseethatwehaveestablishedConjecture 1indimen- sion 1.

3. ThebehaviorofSchneider’snon-convexityindexandtheHausdorffdistance

WeestablishTheorems 3 and4inthissection.Thisreliescruciallyontheelementaryobservationsthatconv(A+^B)= conv(A)+^conv(B)and(t+^s)conv(A)=^tconv(A)+^sconv(A)foranyt,s>0 andanycompactsets A,B.

ProofofTheorem 3. Denote λ=^c(A(k)). Since conv(A(k))=^conv(A), from the definition of c, one knows that A(k)+ λconv(A)=^conv(A)+λconv(A)=(1+λ)conv(A).Usingthat A(k+¹)=_k+^A₁+_k+^k₁^A(k),onehas

A(k+¹)+ ^k

k+¹λconv(A)= ^A k+¹+ ^k

k+^1A(k)+ ^k

k+¹λconv(A)

= ^A k+¹+ ^k

k+^1conv(A)+ ^k

k+¹λconv(A)

⊃^conv(A) k+¹ + ^k

k+^1A(k)+ ^k

k+¹λconv(A)

=^conv(A) k+¹ + ^k

k+¹(1+λ)conv(A)

=

1+ ^k k+1λ

conv(A).

Sincetheotherinclusionistrivial,wededucethat A(k+¹)+_k+^k₁λconv(A)isconvex,whichprovesthat c(A(k+1))≤ ^k

k+¹λ= ^k

k+^1c(A(k)) . 2

ProofofTheorem 4. Letk≥^c(A),then,fromthedefinitionsofc(A)andd(A(k)),onehas conv(A)= ^A

k+¹+ ^k

k+^1conv(A)⊂ ^A k+¹+ ^k

k+¹

A(k)+^d(A(k))Bⁿ₂

=^A(k+¹)+ ^k

k+^1d(A(k))Bⁿ₂.

Weconcludethat d(A(k+¹))≤ ^k

k+^1d(A(k)) . 2

4. Discussion

(i) ByrepeatedapplicationofTheorem 3,itisclearthatthe convergenceofc(A(k))isatarate O(1/k)foranycompact set A⊂^Rn;thisobservationappearstobenew.In[3],westudythequestionofthemonotonicityof A(k),aswell as convergencerates,whenconsidering severaldifferentwaystomeasure non-convexity,includingsome notmentioned inthisnote.

(ii) SomeoftheresultsinthisnoteareofinterestwhenoneisconsideringMinkowskisumsofdifferentcompactsets,not justsumsof Awithcopiesofitself.Indeed,theoriginalconjectureof[1]wasofthisform,andwouldhaveprovideda strengtheningoftheclassicalBrunn–Minkowskiinequalityformorethan2sets;ofcourse,thatconjectureisfalsesince theweakerConjecture 1is false.Nonethelesswe dohavesome relatedobservationsin[3];forinstance,itturns out thatingeneraldimension,forcompactsets A1,. . . ,Ak,

Vol_{n k}

i=¹ A_i

≥ ¹ k−¹

k i=¹

Vol_n

⎛

⎝

j∈[k]\{i} A_j

⎞

⎠.

Forconvexsets Bi,anevenstrongerfactistrue(thatthisisstrongermaynotbe immediatelyobvious),butiffollows fromwell-knownresults,see,e.g.,[5]:

Voln(B1+B2+B3)+Voln(B1)≥Voln(B1+B2)+Voln(B1+B3).

(iii) ThereisavariantofthestrongmonotonicityofSchneider’sindexwhendealingwithdifferentsets.IfA,B,C aresubsets ofRⁿ,thenitisshownin[3](byasimilarargumenttothatusedforTheorem 3)thatc(A+^B+^C)≤^max{^c(A+^B),c(B+ C)}^.

Acknowledgements

Weare indebtedtoFedorNazarovforvaluablediscussions,inparticularforhelpintheconstructionofthecounterex- ample inTheorem 2. Wealso thankFranckBarthe, DarioCordero-Erausquin, UriGrupel, JosephLehec, Bo’azKlartag, and Paul-MarieSamsonforinterestingdiscussions.

References

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Milman(Eds.),Concentration,FunctionalInequalitiesandIsoperimetry,in:Contemp.Math.,vol. 545,Amer.Math.Soc.,2011,pp. 35–53.

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[7]R.M.Starr,Quasi-equilibriainmarketswithnon-convexpreferences,Econometrica37 (1)(1969)25–38.