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The monotonicity of f -vectors of random polytopes
Olivier Devillers, Marc Glisse, Xavier Goaoc, Guillaume Moroz, Matthias
Reitzner
To cite this version:
Olivier Devillers, Marc Glisse, Xavier Goaoc, Guillaume Moroz, Matthias Reitzner. The monotonicity
of f -vectors of random polytopes. Electronic Communications in Probability, Institute of
Mathemat-ical Statistics (IMS), 2013, 18 (23), pp.1-8. �10.1214/ECP.v18-2469�. �hal-00805690�
DOI: 10.1214/ECP.v18-2469
ISSN: 1083-589X COMMUNICATIONSin PROBABILITY
The monotonicity of
f
-vectors of random polytopes
˚Olivier Devillers
:Marc Glisse
;Xavier Goaoc
§Guillaume Moroz
¶Matthias Reitzner
}Abstract
LetKbe a compact convex body inRd, letKnbe the convex hull ofnpoints chosen
uniformly and independently inK, and letfipKnqdenote the number ofi-dimensional
faces ofKn. We show that for planar convex sets,Erf0pKnqsis increasing in n. In
dimensiond ě 3we prove that iflimnÑ8
Erfd´1pKnqs
Anc “ 1for some constantsAand
c ą 0then the functionn ÞÑ Erfd´1pKnqsis increasing fornlarge enough. In
partic-ular, the number of facets of the convex hull ofnrandom points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.
Keywords: random polytopes; convex hull;f-vector. AMS MSC 2010: 60D05;52A22.
Submitted to ECP on November 29, 2012, final version accepted on March 18, 2013.
1
Introduction
What does a random polytope, that is, the convex hull of a finite set of random points inRd, look like? This question goes back to Sylvester’s four point problem, which asked
for the probability that four points chosen at random be in convex position. There are, of course, many ways to distribute points randomly, and as with Sylvester’s problem, the choice of the distribution drastically influences the answer.
Random polytopes. In this paper, we consider a random polytope Kn obtained as
the convex hull ofn points distributed uniformly and independently in a convex body
K Ă Rd, i.e. a compact convex set with nonempty interior. This model arises
natu-rally in applications areas such as computational geometry [7, 11], convex geometry and stochastic geometry [15] or functional analysis [6, 10]. A natural question is to understand the behavior of thef-vector ofKn, that is,f pKnq “ pf0pKnq, . . . , fd´1pKnqq
wherefipKnqcounts the number ofi-dimensional faces, and the behaviour of the volume
˚This research was partially supported by ANR blanc PRESAGE (ANR-11-BS02-003). :Inria Sophia-Antipolis. E-mail: Olivier.Devillers@inria.fr
;Inria Saclay. E-mail: Marc.Glisse@inria.fr
§Université de Lorraine/ CNRS/ Inria, Villers-lès-Nancy, 54600 France.
E-mail: xavier.goaoc@inria.fr
¶Université de Lorraine/ CNRS/ Inria, Villers-lès-Nancy, 54600 France.
E-mail: guillaume.moroz@inria.fr
}Institut für Mathematik, Universität Osnabrück, 49069 Osnabrück, Germany.
The monotonicity off-vectors of random polytopes
V pKnq. Bounding fipKnqis related, for example, to the analysis of the computational
complexity of algorithms in computational geometry.
Exact formulas for the expectation off pKnqand V pKnqare only known for convex
polygons [2, Theorem 5]. In general dimensions, however, the asymptotic behavior, as
ngoes to infinity, is well understood; the general picture that emerges is a dichotomy between the case whereKis a polytope when
ErfipKnqs “ cd,i,Klogd´1n ` oplogd´1nq, (1.1) ErV pKq ´ V pKnqs “ cd,Kn´1logd´1n ` opn´1logd´1nq,
and the case whereK is a smooth convex body (i.e. with boundary of differentiability classC2and with positive Gaussian curvature) when
ErfipKnqs “ c1d,i,Kn d´1 d`1 ` opn d´1 d`1q, (1.2) ErV pKq ´ V pKnqs “ cd,Kn´ 2 d`1 ` opn´ 2 d`1q.
(Herecd,i,Kandc1d,i,K are constants depending only oni,dand certain geometric
func-tionals ofK.) The results forErV pKnqsfollow from the corresponding results forf0pKnq
via Efron’s formula [8]. Numerous papers are devoted to such estimates and we refer to Reitzner [14] for a recent survey.
Monotonicity off-vectors. In spite of numerous papers devoted to the study of ran-dom polytopes, the natural question whether these functionals are monotone remained open in general.
Concerning the monotonicity ofErV pKnqswith respect ton, the elementary
inequal-ity
ErV pKnqs ď ErV pKn`1qs (1.3)
for any fixedKfollows immediately from the monotonicity of the volume. Yet the mono-tonicity (at first sight similarly obvious) with respect toK, i.e. the inequality thatK Ă L
implies
ErV pKnqs ď ErV pLnqs
surprisingly turned out to be false in general. This problem was posed by Meckes [9], see also [14], and a counterexample forn “ d ` 1was recently given by Rademacher [12].
A “tantalizing problem” in stochastic geometry, to quote Vu [16, Section 8], is whether
f0pKnqis monotone inn, that is, whether similar to Equation (1.3): Erf0pKnqs ď Erf0pKn`1qs.
This is not a trivial question as the convex hull ofKnY txuhas fewer vertices thanKn
ifxlies outsideKn and sees more thandof its vertices. Some bounds are known for
the expected number of points of Kn seen by a random point xchosen uniformly in KzKn[16, Corollary 8.3] but they do not suffice to prove thatErf0pKnqsis monotone.
It is known thatErf0pKnqsis always bounded from below by an increasing function
ofn, namely cpdq logd´1n wherecpdq depends only on the dimension: this follows, via Efron’s formula [8], from a similar lower bound on the expected volume ofKn due to
Bárány and Larman [1, Theorem 2]. While this is encouraging, it does not exclude the possibility of small oscillations preventing monotonicity. In fact, Bárány and Larman [1, Theorem 5] also showed that for any functions sand` such thatlimnÑ8spnq “ 0and
ECP 18 (2013), paper 23.
limnÑ8`pnq “ 8there exists1a compact convex domain
K Ă Rdand a sequence pniqiPN
such that for alli P N:
Erf0pKn2iqs ą spn2iqn2i d´1
d`1 and Erf0pKn
2i`1qs ă `pn2i`1q log d´1
pn2i`1q.
When general convex sets are considered, there may thus be more to this monotonicity question than meets the eye.
Results. This paper present two contributions on the monotonicity of thef-vector of
Kn. First, we show that for any planar convex bodyK the expectation off0pKnqis an
increasing function ofn. This is based on an explicit representation of the expectation
Erf0pKnqs.
Theorem 1.1. AssumeKis a planar convex body. For all integersn,
ErfipKn`1qs ą ErfipKnqs, i “ 0, 1.
Second, we show that ifKis a compact convex set inRdwith aC2boundary then the expectation offd´1pKnqis increasing fornlarge enough (where “large enough”depends
onK); in particular, it follows from Euler’s polytope formula and the simplicity ofKn
that for smooth compact convex bodiesKinR3also the expectation off
0pKnqbecomes
monotone innfornlarge enough.
Theorem 1.2. AssumeK Ă Rd is a smooth convex body. Then there is an integern K
such that for alln ě nK
ErfipKn`1qs ą ErfipKnqs, i “ d ´ 2, d ´ 1.
Our result is in fact more general and applies to convex hulls of points i.i.d. from any “sufficiently generic” distribution (see Section 3) and follows from a simple and elegant random sampling technique introduced by Clarkson [4] to analyze non-random geometric structures in discrete and computational geometry.
2
Monotonicity for convex domains in the plane
AssumeK is a planar convex body of volume one. Given a unit vectoru P S1, each
halfspacetx PR2: x ¨ u ď pucuts off fromKa settx P K : x ¨ u ď puof volumes P r0, 1s. On the other hand, givenu P S1, s P p0, 1qthere is a unique linetx PR2: x ¨ u “ puand a corresponding halfspacetx PR2 : x ¨ u ď puwhich cuts off fromK a set of volume
preciselys. Denote byLps, uqthe square of the length of this unique chord.
Using this notation, Buchta and Reitzner [2] showed that the expectation of the number of points on the convex hull ofnpoints chosen randomly uniformly inKcan be computed with the following formula.
Lemma 2.1 (Buchta,Reitzner). Erf0pKnqs “ 1 6npn ´ 1q ż S1 ż1 0 sn´2Lps, uq dsdu
Heredudenotes integration with respect to Lebesgue measure onS1. By a change of variables we obtain Erf0pKnqs “ 1 6pn ´ 1q ż S1 ż1 0 t´1nLptn1, uq dtdu “ 1 6 ż S1 Inpuqdu
The monotonicity off-vectors of random polytopes
with Inpuq “ pn ´ 1q ş1
0t
´1nLptn1, uq dt. Observe that Lp1, uq :“ lim
sÑ1Lps, uq “ 0 for
almost allu P S1. Also observe that for fixed
u P S1the partial derivative B
BsLps, uqexists
for almost alls P p0, 1q. This is a consequence of the a.e. differentiability of a convex function. In the following we writeLp¨q “ Lp¨, uq, B
BsLp¨, uq “ L 1p¨q. Integration by parts intgives Inpuq “ ´ ż1 0 L1ptn1q dt (2.1)
for almost allu P S1. Finally, the convexity ofKinduces the following lemma.
Lemma 2.2. Given a valueu, the derivatives ÞÑ L1psqis a non-increasing function.
Proof. Fixu P S1. We denotelppq “ lpp, uqthe length of the chord
K Xtx P R2: x¨u “ pu,
and define
p “ inftp : lppq ą 0u, p “ suptp : lppq ą 0u
which is the support function ofKin direction ´u, resp. u. Moreover,Lpsppqq “ lppq2
where
sppq “ volptx P K : x ¨ u ď puq
is the volume of the part ofKon the ‘left’ of the chord of lengthlppq. Forp P rp, psthe volumesppq is a monotone and therefore injective function ofpwith inverseppsq, and we have dpdsppq “ lppq.
First observe that becauseKis a convex body, the chord lengthlppqis concave as a function ofp P rp, ps. Thus its derivative dpdlppqis non-increasing.
Hence L1 psq “ 2lpppsqqd dslpppsqq “ 2 d dslpppsqq p d dpsppqq|p“ppsq“ 2 d dplppq|p“ppsq.
Since the derivativedpdlppqis non-increasing andppsqis monotone, we see thatdsdLps, uq
is non-increasing ins.
This allows us to conclude that the expectancy of the number of points on the convex hull is increasing.
Proof of Theorem 1.1. According to Lemma 2.2, L1psq is decreasing. Since for allt P r0, 1s,tn1 ă t
1
n`1, this implies that L1pt1nq ě L1pt 1
n`1q. Combined with equation (2.1) we
have
Inpuq ď In`1puq
which provesErf0pKnqs ď Erf0pKn`1qs. In the planar case, the number of edges is also
an increasing function becausef0pKnq “ f1pKnq.
3
Random sampling
We denote by 7S the cardinality of a finite setS and we let1X denote the
charac-teristic function of eventX: 1pPF is 1 if p P F and0 otherwise. Let S be a finite set of points inRd and letk ě 0 be an integer. Ak-set ofS is a subset
tp1, p2, . . . pdu Ď S
that spans a hyperplane bounding an open halfspace that contains exactlykpoints from
S; we say that thek-set cuts off thesek points. In particular, ifS is a set of points in general position, its0-sets are the facets of its convex hull, which we denote CHpSq. For any finite subsetSofDwe letskpSqdenote the number2ofk-sets ofS.
2If the hyperplane separatesSin two subsets ofkelements (2k ` d “ n) thek-set is counted twice.
ECP 18 (2013), paper 23.
Generic distribution assumption. LetP denote a probability distribution onRd; we
assume throughout this section thatP is such thatdpoints chosen independently from
P are generically affinely independent.
Determining the order of magnitude of the maximum number of k-sets determined by a set ofnpoints inRdhas been an important open problem in discrete and computa-tional geometry over the last decades. In the cased “ 2, Clarkson [4] gave an elegant proof thatsďkpSq “ Opnkqfor any setSofnpoints in the plane, where:
sďkpSq “ s0pSq ` s1pSq ` . . . ` skpSq.
(See also Clarkson and Shor [5] and Chazelle [3, Appendix A.2].) Although the state-ment holds for any fixed point set, and not only in expectation, Clarkson’s argustate-ment is probabilistic and will be our main ingredient. It goes as follows. LetRbe a subset ofS
of sizer, chosen randomly and uniformly among all such subsets. Ani-set ofSbecomes a0-set inRifRcontains the two points defining thei-set but none of theipoints it cuts off. This happens approximately with probabilityp2
p1 ´ pqi, wherep “ nr (see [5] for exact computations). It follows that:
Ers0pRqs ě ÿ
0ďiďk
p2p1 ´ pqisipSq ě p2p1 ´ pqksďkpSq.
SinceErs0pRqs cannot exceed 7R “ r, we have sďkpSq ď pp1´pqn k which, forp “ 1{k,
yields the announced bound sďkpSq “ Opnkq. A similar random sampling argument
yields the following inequalities.
Lemma 3.1. Letskpnqdenote the expected number ofk-sets in a set ofnpoints chosen
independently fromP. We have
s0pnq“s0pn ´ 1q `
d s0pnq ´ s1pnq
n (3.1)
and for any integer1 ď r ď n
s0prq ě `n´d r´d ˘ `n r ˘ s0pnq ` `n´d´1 r´d ˘ `n r ˘ s1pnq. (3.2)
Proof. LetSbe a set ofnpoints chosen, independently, fromP and letq P S. The0-sets ofSthat are not0-sets ofSztquare precisely those that containq. Conversely, the0-sets ofSztquthat are not0-sets ofS are precisely those1-sets of S that cut offq. We can thus write:
s0pSq “ s0pSztquq `
ÿ
F facet ofCHpSq
1qPF ´ 71-sets cutting offq.
Note that the equality remains true in the degenerate cases where several points ofS
are identical or in a non generic position if we count the facets of the convex hull of
S with multiplicities in the sum and in s0pnq. Summing the previous identity over all
pointsqofS, we obtain ns0pSq “ ˜ ÿ qPS s0pSztquq ¸ ` ¨ ˝ ÿ F facet ofCHpSq ÿ qPS 1qPF ˛ ‚´ s1pSq,
The monotonicity off-vectors of random polytopes ns0pSq“ ˜ ÿ qPS s0pSztquq ¸ ` ds0pSq ´ s1pSq almost surely.
Takingnpoints chosen randomly and independently fromP, then deleting one of these points, chosen randomly with equiprobability, is the same as takingn ´ 1chosen ran-domly and independently fromP. We can thus average over all choices ofSand obtain
ns0pnq“ns0pn ´ 1q ` d s0pnq ´ s1pnq,
which implies Equality (3.1).
Now, letr ď nand letRbe a random subset ofS of sizer, chosen uniformly among all such subsets. For k ď r
2, ak-set of S is a 0-set of R if R contains that k-set and
none of thekpoints it cuts off. For each fixedk-setAofS, there are therefore`n´d´kr´d ˘ distinct choices ofRin whichAis a0-set. Counting the expected number of0-sets and
1-sets of S that remain/become a0-set inR, and ignoring those0-sets of Rthat were
k-sets ofS fork ě 2, we obtain:
Ers0pRqs ě `n´d r´d ˘ `n r ˘ s0pSq ` `n´d´1 r´d ˘ `n r ˘ s1pSq.
Recall that the expectation is taken over all choices of a subsetRofrelements of the fixed point set S. We can now average over all choices of a setS of n points taken, independently, fromP, and obtain Inequality (3.2).
Lettingp “ n´dr´d, Inequality (3.2) yields the, perhaps more intuitive, inequality:
s0prq ě pds0pnq ` pdp1 ´ pqs1pnq.
We can now prove our main result.
Theorem 3.2. LetZndenote the convex hull ofnpoints chosen independently fromP.
IfErfd´1pZnqs „ Anc whenn Ñ 8, for someA, c ą 0, then there exists an integern0
such that for anyn ě n0we haveErfd´1pZn`1qs ą Erfd´1pZnqs.
Proof of Theorem 3.2. Let P be a probability distribution on Rd and let s
kpnqdenote
the expected number ofk-sets in a set ofnpoints chosen independently fromP. Recall that s0 counts the expected number of facets in the convex hull of n points chosen
independently and uniformly fromP. By Equality (3.1), fors0to be increasing it suffices
thats1pnqbe bounded from above byds0pnq.
Letψpr, nq “ s0prq
s0pnq. Substituting into Inequality (3.2), ψpr, nqs0pnq ě `n´d r´d ˘ `n r ˘ s0pnq ` `n´d´1 r´d ˘ `n r ˘ s1pnq, which rewrites as s1pnq ď `n´d r´d ˘ `n´d´1 r´d ˘ ˜ ψpr, nq `n r ˘ `n´d r´d ˘ ´ 1 ¸ s0pnq (3.3)
We letq “ n´dr´d. Developing the binomial expressions:
`n´d r´d ˘ `n´d´1 r´d ˘ “ n ´ d n ´ r “ q q ´ 1 and `n r ˘ `n´d r´d ˘ “ n r ¨ n ´ 1 r ´ 1. . . n ´ d ` 1 r ´ d ` 1 ECP 18 (2013), paper 23. Page 6/8 ecp.ejpecp.org
And for0 ď k ă dwe have n´k r´k ă n´d r´d “ q. Thus: s1pnq ď q ψpr, nqnrqd´1´ 1 q ´ 1 s0pnq
Assume now that we know a functiongsuch thats0pnq „ gpnq. Then for any 12ą ą 0, there isNPNsuch that for alln ą r ą Nwe have:
ψpr, nq “ s0prq s0pnq ăˆ 1 ` 1 ´ ˙ gprq gpnq ă p1 ` 4q gprq gpnq which gives: s1pnq ď q gprq{r gpnq{nq d´1´ 1 q ´ 1 s0pnq ` 4 ˆ gprq{r gpnq{n ˙ qd q ´ 1s0pnq. (3.4)
In the case wheres0pnq „ Anc, we have: gprq{r gpnq{n“ ´n r ¯1´c ă q1´c
And plugging back in Equation (3.4), we get:
s1pnq ď q qd´c´ 1 q ´ 1 s0pnq ` 4 qd`1 q ´ 1s0pnq (3.5) The expression qqd´c´1
q´1 converges toward d ´ cwhenq approaches1. Thus, there
existsdsuch that for all1 ă q ă 1 ` d:
qq d´c´ 1
q ´ 1 ă d ´ c 2
The second term of Equation (3.5) is bounded for all1 `d
2 ă q ă 1 ` dby: 4q d`1 q ´ 1s0pnq ă 8 p1 ` dqd`1 d s0pnq Finally, let “ cd
32p1`dqd`1. For allrsuch that:
Nă r ă n and 1 ` d 2 ă n ´ d r ´ d ă d (3.6) we have: s1pnq ă pd ´ c 4qs0pnq
With Equality (3.1) this implies thats0pnq ą s0pn ´ 1q.
It remains to check that fornlarge enough, there always existsrsatisfying Condition (3.6). We can rewrite condition (3.6) as:
d ` n ´ d 1 ` d
ă r ă d ` n ´ d 1 ` d{2
In particular, there exists an integerrsatisfying this condition as soon as1`n´d
d{2´ n´d 1`d ą 1. Thus, as soon as n ą maxpN, d ` 1 1
1`d{2´ 1 1`d
q, Condition (3.6) is satisfied, which concludes the proof.
The monotonicity off-vectors of random polytopes
Now, from equations (1.1) and (1.2) we can see that Theorem 3.2 holds for random polytopesKn when K is smooth, but not when K is a polytope. This proves that for
smoothKthe expectationErfd´1pKnqsis asymptotically increasing, i.e. the first part of
Theorem 1.2.
The genericity assumption on P implies that the convex hullZn ofnpoints chosen
independently fromP is almost surely simplicial. Thus, inZn anypd ´ 1q-face is almost
surely incident to exactly dfaces of dimensiond ´ 2and fd´2pZnq “ d2fd´1pZnq.
The-orem 3.2 therefore implies thatfd´2pZnq is asymptotically increasing, i.e. the second
part of Theorem 1.2.. Ford “ 3, with Euler’s relation this further implies thatf0pZnqis
asymptotically increasing.
Acknowledgments. This work was initiated during the10thMcGill - INRIA Workshop on Computational Geometry at the Bellairs institute. The authors wish to thank all the participants for creating a pleasant and stimulating atmosphere, in particular Sariel Har-Peled and Raimund Seidel for useful discussions at the early stage of this work. The authors also thank Imre Bárány and Pierre Calka for helpful discussions.
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ECP 18 (2013), paper 23.
