The monotonicity of f-vectors of random polytopes

Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is increasing in n. In dimension d>=3 we prove that if lim(E((f[d -1](Kn))/(An^c)->1 when n->infinity for some constants A and c>0 then the function E(f[d-1](Kn)) is increasing for n large enough. In particular, the number of facets of the convex hull of n random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.


Résumé :
Soit K un domaine convexe et compact de R d , K n l'enveloppe convexe de n points choisis uniformément et indépendamment dans K, et f i pK n q le nombre de faces de K n de dimension i.

Introduction
What does a random polytope, that is, the convex hull of a finite set of random points in R d , 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 K n obtained as the convex hull of n points distributed uniformly and independently in a convex body K Ă R d , i.e. a compact convex set with nonempty interior.This model arises naturally 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 the f -vector of K n , that is, f pK n q " pf 0 pK n q, . . ., f d´1 pK n qq where f i pK n q counts the number of i-dimensional faces, and the behaviour of the volume V pK n q .Bounding f i pK n q is related, for example, to the analysis of the computational complexity of algorithms in computational geometry.
Exact formulas for the expectation of f pK n q and V pK n q are only known for convex polygons [2, Theorem 5].In general dimensions, however, the asymptotic behavior, as n goes to infinity, is well understood; the general picture that emerges is a dichotomy between the case where K is a polytope when ErV pKq ´V pK n qs " c d,K n ´1 log d´1 n `opn ´1 log d´1 nq, and the case where K is a smooth convex body (i.e. with boundary of differentiability class C 2 and with positive Gaussian curvature) when ErV pKq ´V pK n qs " c d,K n ´2 d`1 `opn ´2 d`1 q.
(Here c d,i,K and c 1 d,i,K are constants depending only on i, d and certain geometric functionals of K.) The results for ErV pK n qs follow from the corresponding results for f 0 pK n q via Efron's formula [8].The literature devoted to such estimates is consequent and we refer to Reitzner [14] for a recent survey.
Monotonicity of f -vectors.In spite of numerous papers devoted to the study of random polytopes, the natural question whether these functionals are monotone remained open in general.
Concerning the monotonicity of ErV pK n qs with respect to n, the elementary inequality for any fixed K follows immediately from the monotonicity of the volume.Yet the monotonicity (at first sight similarly obvious) with respect to K, i.e. the inequality that K Ă L implies ErV pK n qs ď ErV pL n qs surprisingly turned out to be false in general.This problem was posed by Meckes [9], see also [14], and a counterexample for n " d `1 was recently given by Rademacher [12].
A "tantalizing problem" in stochastic geometry, to quote Vu [16,Section 8], is whether f 0 pK n q is monotone in n, that is, whether similar to Equation (3): Erf 0 pK n qs ď Erf 0 pK n`1 qs.This is not a trivial question as the convex hull of K n Y txu has fewer vertices than K n if x lies outside K n and sees more than two of its vertices.Some bounds are known for the expected number of points of K n seen by a random point x chosen uniformly in KzK n [16,Corollary 8.3] but they do not suffice to prove that Erf 0 pK n qs is monotone.
It is known that Erf 0 pK n qs is always bounded from below by an increasing function of n, namely cpdq log d´1 n where cpdq depends only on the dimension: this follows, via Efron's formula [8], from a similar lower bound on the expected volume of K n 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 s and ℓ such that lim nÑ8 spnq " 0 and lim nÑ8 ℓpnq " 8 there exists1 a compact convex domain K Ă R d and a sequence pn i q iPN such that for all i P N: and Erf 0 pK n2i`1 qs ă ℓpn 2i`1 q log d´1 pn 2i`1 q.
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 the f -vector of K n .First, we show that for any planar convex body K the expectation of f 0 pK n q is an increasing function of n.This is based on an explicit representation of the expectation Erf 0 pK n qs.Theorem 1. Assume K is a planar convex body.For all integers n, Erf i pK n`1 qs ą Erf i pK n qs, i " 0, 1.
Second, we show that if K is a compact convex set in R d with a C 2 boundary then the expectation of f d´1 pK n q is increasing for n large enough (where "large enough"depends on K); in particular, for smooth compact convex bodies K in R d the expectation of f 0 pK n q becomes monotone in n for n large enough.
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.

Monotonicity for convex domains in the plane
Assume K is a planar convex body of volume one.Given a unit vector u P S 1 , each halfspace tx P R 2 : x ¨u ď pu cuts off from K a set of volume s P r0, 1s.On the contrary, given u P S 1 , s P p0, 1q there is a unique line tx P R 2 : x ¨u " pu and a corresponding halfspace tx P R 2 : x ¨u ď pu which cuts of from K a set of volume precisely s.Denote by Lps, uq the 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 of n points chosen randomly uniformly in K can be computed with the following formula.
Here du denotes integration with respect to Lebesgue measure on S 1 .By a change of variables we obtain I n puqdu with I n puq " pn ´1q ş 1 0 t ´1 n Lpt 1 n , uq dt.Observe that Lp1, uq " 0 for almost all u P S 1 .Also observe that the partial derivative B Bs Lps, uq exists for almost all ps, uq.This is a consequence of the a.e.differentiability of a convex function.In the following we write Lp¨q " Lp¨, uq, B Bs Lp¨, uq " L 1 p¨q.Integration by parts in t gives Finally, the convexity of K induces the following lemma.
Lemma 4. Given a value u, the derivative s Þ Ñ L 1 psq is a decreasing function.
Proof.Fix u P S 1 .We denote lppq " lpp, uq the length of the chord K X tx P R 2 : x ¨u " pu.Moreover, Lpsppqq " lppq 2 where sppq is the volume of the part of K on the left of the chord of length lppq.This volume sppq is a monotone and hence injective function of p with inverse ppsq, and we have d dp sppq " lppq.First observe that because K is a convex body, the chord length lppq is concave as a function of p.This allows us to conclude that the expectancy of the number of points on the convex hull is increasing.
Proof of Theorem 1.According to Lemma 4, L 1 psq is decreasing.Since for all t P r0, 1s, t Combined with equation (4) we have I n puq ď I n`1 puq which proves Erf 0 pK n qs ď Erf 0 pK n`1 qs.In the planar case, the number of edges is also an increasing function because f 0 pK n q " f 1 pK n q.

Random sampling
We denote by 7S the cardinality of a finite set S and we let 1 X denote the characteristic function of event X: 1 pPF is 1 if p P F and 0 otherwise.Let S be a finite set of points in R d and let k ě 0 be an integer.A k-set of S is a subset tp 1 , p 2 , . . .p d u Ď S that spans a hyperplane bounding an open halfspace that contains exactly k points from S; we say that the k-set cuts off these k points.In particular, 0-sets are facets of the convex hull of S. For any finite subset S of D we let s k pSq denote the number2 of k-sets of S. Generic distribution assumption.Let P denote a probability distribution on R d ; we assume throughout this section that P is such that d points chosen independently from P are generically affinely independent.
Determining the order of magnitude of the maximum number of k-sets determined by a set of n points in R d has been an important open problem in discrete and computational geometry over the last decades.In the case d=2, Clarkson [4] gave an elegant proof that s ďk pSq " Opnkq for any set S of n points in the plane, where: s ďk pSq " s 0 pSq `s1 pSq `. . .`sk pSq.
(See also Clarkson and Shor [5] and Chazelle [3,Appendix A.2].)Although the statement holds for any fixed point set, and not only in expectation, Clarkson's argument is probabilistic and will be our main ingredient.It goes as follows.Let R be a subset of S of size r, chosen randomly and uniformly among all such subsets.A i-set of S becomes a 0-set in R if R contains the two points defining the i-set but none of the i points it cuts off.This happens approximately with probability p 2 p1 ´pq i , where p " r n (see [5] for exact computations).It follows that: Ers 0 pRqs ě ÿ 0ďiďk p 2 p1 ´pq i s i pSq ě p 2 p1 ´pq k s ďk pSq.
Since Ers 0 pRqs cannot exceed 7R " r, we have s ďk pSq ď n pp1´pq k which, for p " 1{k, yields the announced bound s ďk pSq " Opnkq.A similar random sampling argument yields the following inequalities.
Proof.Let S be a set of n points chosen, independently, from P and let q P S. The 0-sets of S that are not 0-sets of Sztqu are precisely those defined by q.Conversely, the 0-sets of Sztqu that are not 0-sets of S are precisely those 1-sets of S that cut off q.We can thus write: Note that the equality remains true in the degenerate cases where several points of S 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 s 0 pnq.Summing the previous identity over all points q of S, we obtain This inequality is actually an equality if d " 2 but not if d ě 3. Taking n points chosen randomly and independently from P, then deleting one of these points, chosen randomly with equiprobability, is the same as taking n ´1 chosen randomly independently from P. We can thus average over all choices of S and obtain ns 0 pnq ě ns 0 pn ´1q `d s 0 pnq ´s1 pnq, which implies Inequality (5).Now, let r ď n and let R be a random subset of S of size r, chosen uniformly among all such subsets.For k ď r 2 , a k-set of S is a 0-set of R if R contains that k-set and none of the k points it cuts off.For each fixed k-set A of S, there are therefore `n´d´k r´d ˘distinct choices of R in which A is a 0-set.Counting the expected number of 0-sets and 1-sets of S that remain/become a 0-set in R, and ignoring those 0-sets of R that were k-sets of S for k ě 2, we obtain: Recall that the expectation is taken over all choices of a subset R of r elements of the fixed point set S. We can now average over all choices of a set S of n points taken, independently, from P, and obtain Inequality (6).
We can now prove our main result.
Theorem 6.Let Z n denote the convex hull of n points chosen independently from P. If Erf d´1 pZ n qs « An c for some A, c ą 0 then there exists an integer n 0 such that for any n ě n 0 we have Erf d´1 pZ n`1 qs ą Erf d´1 pZ n qs.
Proof of Theorem 6.Let P be a probability distribution on R d and let s k pnq denote the expected number of k-sets in a set of n points chosen independently from P. Recall that s 0 counts the expected number of facets in the convex hull of n points chosen independently and uniformly from P. By Inequality (5), for s 0 to be increasing it suffices that s 1 pnq be bounded from above by ds 0 pnq.
We let q " n´d r´d .Developing the binomial expressions: And for 0 ď k ă d we have n´k r´k ă n´d r´d " q.Thus: Assume now that we know a function g such that s 0 pnq « gpnq.Then for any 1 2 ą ǫ ą 0, there is N ǫ P N such that for all n ą r ą N ǫ we have: ´ǫ ˙gprq gpnq ă p1 `4ǫq gprq gpnq which gives: In the case where s 0 pnq « An c , we have: And plugging back in Equation ( 8), we get: The expression q q d´c ´1 q´1 converges toward d ´c when q approaches 1.Thus, there exists ǫ d such that for all 1 ă q ă 1 `ǫd : The second term of Equation ( 9) is bounded for all 1 `ǫd 2 ă q ă 1 `ǫd by: 4ǫ q d`1 q ´1 s 0 pnq ă 8ǫ p1 `ǫd q d`1 ǫ d s 0 pnq Finally, let ǫ " cǫ d 32p1`ǫ d q d`1 .For all r such that: With Inequality (5) this implies that s 0 pnq ą s 0 pn ´1q.It remains to check that for n large enough, there always exists r satisfying Condition (10).We can rewrite condition (10) as: In particular, there exists an integer r satisfying this condition as soon as n´d 1`ǫ d {2 ´n´d 1`ǫ d ą 1.Thus, as soon as n ą maxpN ǫ , d `1 1 1`ǫ d {2 ´1 1`ǫ d q, Condition (10) is satisfied, which concludes the proof.Now, from equations ( 1) and ( 2) we can see that Theorem 6 holds for random polytopes K n when K is smooth, but not when K is a polytope.This proves that for smooth K the expectation Erf d´1 pK n qs is asymptotically increasing, i.e. the first part of Theorem 2.
The genericity assumption on P implies that the convex hull Z n of n points chosen independently from P is almost surely simplicial.Thus, in Z n any pd ´1q-face is almost surely incident to exactly d faces of dimension d ´2 and f d´2 pZ n q " d 2 f d´1 pZ n q.Theorem 6 therefore implies that f d´2 pZ n q is asymptotically increasing, i.e. the second part of Theorem 2..For d " 3, with Euler's relation this further implies that f 0 pZ n q is asymptotically increasing.

Lemma 5 . 5 ) and for any integer 1 ď r ď n s 0
Let s k pnq denote the expected number of k-sets in a set of n points chosen independently from P. We have s 0 pnq ě s 0 pn ´1q `d s 0 pnq ´s1 pnq n (