A note on volume thresholds for random polytopes
Debsoumya Chakraborti, Tomasz Tkocz, Beatrice-Helen Vritsiou
aa r X i v : . [ m a t h . M G ] A p r A note on volume thresholds for random polytopes
Debsoumya Chakraborti ∗ Tomasz Tkocz † Beatrice-Helen Vritsiou ‡ Abstract
We study the expected volume of random polytopes generated by taking theconvex hull of independent identically distributed points from a given distribution.We show that for log-concave distributions supported on convex bodies, we needat least exponentially many (in dimension) samples for the expected volume to besignificant and that super-exponentially many samples suffice for concave measureswhen their parameter of concavity is positive.
Primary 52A23; Secondary 52A22, 60D05;
Key words. random polytopes, convex bodies, log-concave measures, volume threshold, high dimen-sions.
Let X , X , . . . be independent identically distributed (i.i.d.) random vectors uniformon a set K in R n . Let K N = conv { X , . . . , X N } . (1)We are interested in bounds on the number N of points needed for the volume | K N | of K N to be asymptotic in expectation to the volume | conv K | of the convex hull of K as n → ∞ . In the pioneering work [12], Dyer, F¨uredi and McDiarmid establishedsharp thresholds for the vertices of the cube, K = {− , } n as well as for the solid cube K = [ − , n . More precisely, they showed that for every ε > E | K N | n −−−−→ n →∞ ( , if N ≤ ( ν − ε ) n , , if N ≥ ( ν + ε ) n , (2)where for K = {− , } n , we have ν = 2 / √ e = 1 . ... and for K = [ − , n , we have ν = 2 πe − γ − / = 2 . ... (see also [13]). For further generalisations establishing sharpexponential thresholds see [16] (in a situation when the X i are not uniform on a set buthave i.i.d. components compactly supported in an interval).The case of a Euclidean ball is different. Pivovarov showed in [22] (see also [7]) thatwhen K = B n { x ∈ R n , X x i ≤ } , the threshold is superexponential, that is for every ε > E | K N || K | −−−−→ n →∞ ( , if N ≤ e (1 − ε ) · n log n , , if N ≥ e (1+ ε ) · n log n . (3) ∗ Carnegie Mellon University; Pittsburgh, PA 15213, USA. Email: [email protected]. † Carnegie Mellon University; Pittsburgh, PA 15213, USA. Email: [email protected]. Researchsupported in part by the Collaboration Grants from the Simons Foundation. ‡ University of Alberta in Edmonton, Canada. Email: [email protected].
1e additionally considered the situation when the X i are not uniform on a set but areGaussian.In recent works [7, 8], the authors study the case of the X i having rotationallyinvariant densities of the form (1 − P x i ) β B n , β > −
1. This is the so-called Betamodel of random polytopes attracting considerable attention in stochastic geometry.In particular, β = 0 corresponds to the uniform distribution on the unit ball and thelimiting case β → − ε ∈ (0 ,
1) andsequences N = N ( n ), − < β = β ( n ), we have E | K N || B n | −−−−→ n →∞ ( , if N ≤ e (1 − ε )( n + β ) log n , , if N ≥ e (1+ ε )( n + β ) log n , (4)which was further refined in [8]: for every positive constant c , the limit is e − c if N growslike e ( n + β ) log n c as n → ∞ .We would like to focus on establishing general bounds for some large natural familiesof distributions. Specifically, suppose that for each dimension n , we are given a family { µ n,i } i ∈ I n of probability measures such that each µ n,i is compactly supported on acompact set V n,i in R n . We would like to find the largest number N and the smallestnumber N (in terms of n and some parameters of the family) such that for every µ n,i from the family, E | K N || conv V n,i | = o (1) for N ≤ N and E | K N || conv V n,i | = 1 − o (1) for N ≥ N as n → ∞ ( K N is a random polytope given by (1) with X , X , . . . being i.i.d. drawn from µ n,i ).For instance, the examples of the cube and the ball suggest that for the family ofuniform measures on convex bodies, N is exponential and N is super-exponential in n .In fact, the latter can be quickly deduced from a classical result by Groemer from[17], combined with the thresholds for Euclidean balls established by Pivovarov in [22].Groemer’s theorem says that for every N > n , we have E | conv { X , . . . , X N }| ≥ E | conv { Y , . . . , Y N }| , where the X i are i.i.d. uniform on a convex set K and the Y i are i.i.d. uniform on aEuclidean ball with the same volume as K . We thus get from (3) that E | conv { X , . . . , X N }| = 1 − o (1) , (5)as long as N ≥ e (1+ ε ) n log n .In this work, we shall establish an exponential bound on N for the family oflog-concave distributions on convex sets and extend (5) to the family of the so-called κ -concave distributions. Acknowledgements.
We would like to thank Alan Frieze for many helpful discus-sions.
Recall that a Borel probability measure µ on R n is κ -concave, κ ∈ [ −∞ , n ], if for every λ ∈ [0 ,
1] and every Borel sets A , B in R n , we have µ ( λA + (1 − λ ) B ) ≥ (cid:16) λµ ( A ) κ + (1 − λ ) µ ( B ) κ (cid:17) /κ . We say that a random vector is κ -concave if its law is κ -concave. For example, vectorsuniform on convex bodies in R n are 1 /n -concave. The right hand side increases with2 , so as κ increases, the class of κ -concave measures becomes smaller. It is a naturalextension of the class of log-concave random vectors, corresponding to κ = 0, with theright hand side in the defining inequality understood as the limit κ → κ ∈ (0 , /n ). Then a κ -concave random vector is supported on a convex bodyand its density is a 1 /β -concave function, that is of the form h β for a concave function h and β = κ − − n . The notion of κ -concavity was introduced and studied by Borellin [9, 10], which are standard references on this topic. We also recall that a randomvector X in R n is isotropic if it is centred, that is E X = 0 and its covariance matrixCov( X ) = [ E X i X j ] i,j ≤ n is the identity matrix. The isotropic constant L X of a log-concave random vector X with density f is then defined as L X = (ess sup R n f ) /n (see,e.g. [11]).Our first main result concerns an exponential lower bound for the family of symmetriclog-concave distributions supported in convex bodies. Theorem 1.
Let µ be a symmetric log-concave probability measure supported on a con-vex body K in R n . Let X , X , . . . be i.i.d. random vectors distributed according to µ .Let K N = conv { X , . . . , X N } . There are universal positive constants c , c such that if N ≤ e c n/L µ , then E | K N || K | ≤ e − c n/L µ , where L µ is the isotropic constant of µ . Our second main result provides a super-exponential upper bound for the family of κ -concave distributions. Theorem 2.
Let µ be a symmetric κ -concave measure on R n with κ ∈ (0 , n ) , supportedon a convex body K in R n . Let X , X , . . . be i.i.d. random vectors uniformly distributedaccording to µ . Let K N = conv { X , . . . , X N } . There is a universal constant C suchthat for every ω > C , if N ≥ e κ (log n +2 log ω ) , then E | K N || K | ≥ − ω . It turns out that the following quasi-concave function plays a crucial role in estimates forthe expected volume of the convex hull of random points (see [2, 3, 12]): for a randomvector X in R n define q X ( x ) = inf { P ( X ∈ H ) , H half-space containing x } , x ∈ R n . (6)It is clear that q ( λx + (1 − λ ) y ) ≥ min { q ( x ) , q ( y ) } , because if a half-space H contains λx + (1 − λ ) y , it also contains x or y . Consequently, superlevel sets L q X ,δ = { x ∈ R n , q X ( x ) ≥ δ } (7)of this function are convex. Another way of looking at these sets is by noting that theyare intersections of half-spaces: L q X ,δ = T { H : H is a half-space , P ( X ∈ H ) > − δ } .When X is uniform on a convex set K , they are called convex floating bodies ( K \ L q X ,δ is called a wet part). The function q X in statistics is called the Tukey or half-spacedepth of X . The two notions have been recently surveyed in [21].A key lemma from [12] relates the volume of random convex hulls of i.i.d. samples of X to the volume the level sets L q X ,δ . Bounds on the latter are obtained by a combination3f elementary convexity arguments and deep results from asymptotic convex geometry(notably, Paouris’ reversal of the L p -affine isoperimetric inequality due to Lutwak, Yangand Zhang). We shall present these and all the necessary background material in Section4. Section 5 is devoted to our proofs. κ -concave measures Theorem 4.3 from [10] provides in particular the following stability of κ -concavity withrespect to taking marginals: if κ ∈ (0 , n ) and f is the density of a κ -concave randomvector in R n , thenthe marginal x Z R n − f ( x, y )d y is a κ − κ -concave function . (8)We will also need the following basic estimate: if g : R → [0 , + ∞ ) is the density of alog-concave random variable X with E X = 0 and E X = 1, then12 √ e ≤ g (0) ≤ √ The following is a key lemma from [12] (called by the authors “central”) about asymp-totically matching upper and lower bounds for the volume of the random convex hull.
Lemma 3 ([12]) . Suppose X , X , . . . are i.i.d. continuous random vectors in R n . Let K N = conv { X , . . . , X N } and define q = q X by (6) . Then for every subset A of R n ,we have E | K N | ≤ | A | + N · (cid:18) sup A c q (cid:19) · | A c ∩ { x ∈ R n , q ( x ) > }| (10) and E | K N | ≥ | A | (cid:18) − (cid:18) Nn (cid:19) (cid:16) − inf A q (cid:17) N − n (cid:19) . (11)(The proof therein concerns only the cube, but their argument repeated verbatimjustifies our general situation as well – see also [16]). q Lemma 3 is applied to level sets L q,δ of the function q (see (7)). We gather here severalremarks concerning bounds for the volume of such sets. For the upper bound, we willneed the containment L q,δ ⊂ cZ α ( X ), where c is a universal constant and Z α is thecentroid body (defined below). This was perhaps first observed in Theorem 2.2 in [28](with a reverse inclusion as well). We recall an argument below. Remark . Plainly, for the infimum in the definition (6) of q X ( x ), it is enough to takehalf-spaces for which x is on the boundary, that is q X ( x ) = inf θ ∈ R n P ( h X − x, θ i ≥ , (12)4here h u, v i = P i u i v i is the standard scalar product in R n . Of course, by homogeneity,this infimum can be taken only over unit vectors. We also remark that by Chebyshev’sinequality, P ( h X − x, θ i ≥ ≤ e −h θ,x i E e h θ,X i . Consequently, q X ( x ) ≤ exp (cid:18) − sup θ ∈ R n n h θ, x i − log E e h θ,X i o(cid:19) and we have arrived at the Legendre transform Λ ⋆X of the log-moment generating func-tion Λ X of X ,Λ X ( x ) = log E e h X,x i and Λ ⋆X ( x ) = sup θ ∈ R n {h θ, x i − Λ X ( θ ) } . Thus, for every α >
0, we have { x ∈ R n , q X ( x ) > e − α } ⊂ { x ∈ R n , Λ ⋆X ( x ) < α } . (13) Remark . The level sets { Λ ⋆X < α } have appeared in a different context of the so-called optimal concentration inequalities introduced by Lata la and Wojtaszczyk in [19].Modulo universal constants, they turn out to be equivalent to centroid bodies playinga major role in asymptotic convex geometry (see [20, 23, 24, 25, 26]). Specifically, for arandom vector X in R n and α ≥
1, we define its L α -centroid body Z α ( X ) by Z α ( X ) = { x ∈ R n , sup {h x, θ i , E | h X, θ i | α ≤ } ≤ } (equivalently, the support function of Z α ( X ) is θ ( E | h X, θ i | α ) /α ). By Propositions3.5 and 3.8 from [19], if X is a symmetric log-concave random vector X (in particular,uniform on a symmetric convex body), { Λ ∗ X < α } ⊂ eZ α ( X ) , α ≥ . (14)(A reverse inclusion Z α ( X ) ⊂ /α e { Λ ∗ X < α } holds for any symmetric random vector,see Proposition 3.2 therein.)We shall need an upper bound for the volume of centroid bodies. This was done byPaouris (see [25]). Specifically, Theorem 5.1.17 from [11] says that if X is an isotropiclog-concave random vector in R n , then | Z α ( X ) | /n ≤ C r αn , ≤ α ≤ n, (15)where C is a universal constant. Remark . Significant amount of work in [12] was devoted to showing that for thecube inclusion (13) is nearly tight (for correct values of α , using exponential tilting ofmeasures typically involved in establishing large deviation principles). We shall take adifferent route and put a direct lower bound on q X described in the following lemma.Our argument is based on property (8). Lemma 7.
Let κ ∈ (0 , n ) . Let X be a symmetric isotropic κ -concave random vectorsupported on a convex body K in R n . Then for every unit vector θ in R n and a > , wehave P ( h X, θ i > a ) ≥ κ (cid:18) − ah K ( θ ) (cid:19) /κ , (16) where h K ( θ ) = sup y ∈ K h y, θ i is the support function of K . In particular, denoting thenorm given by K as k · k K , we have q X ( x ) ≥ κ (1 − k x k K ) /κ , x ∈ K. (17)5 roof. Consider the density g of h X, θ i . Let b = h K ( θ ). Note that g is supported in[ − b, b ]. By (8), g κ − κ is concave, thus on [0 , b ] we can lower-bound it by a linear functionwhose values agree at the end points, g ( t ) κ − κ ≥ g (0) κ − κ (cid:18) − tb (cid:19) , t ∈ [0 , b ] . This gives P ( h X, θ i > a ) = Z ba g ( t )d t ≥ g (0) Z ba (cid:18) − tb (cid:19) − κκ d t = κg (0) b (cid:16) − ab (cid:17) /κ . Since h X, θ i is in particular log-concave, by (9), we have √ e ≤ g (0) ≤ √
2. Moreover,by isotropicity, 1 = E h X, θ i = Z b − b t g ( t )d t ≤ b g (0) . Thus, say g (0) b > and we get (16). To see (17), first recall (12). By symmetry, P ( h X − x, θ i ≥
0) = P ( h X, θ i ≥ | h x, θ i | ), so we use (16) with a = | h θ, x i | and notethat by the definition of h K , | D x k x k K , θ E | ≤ h K ( θ ), so |h x,θ i| h K ( θ ) ≤ k x k K . Since the quantity E | K N || K | does not change under invertible linear transformations appliedto µ , without loss of generality we can assume that µ is isotropic. Let q = q X be definedby (6). Fix α > A = { x, q ( x ) > e − α } . We get E | K N || K | ≤ | A || K | + N e − α (we have used { x, q ( x ) > } ⊂ K to estimate the last factor in (10) by 1). Combining(13), (14) and (15), | A | ≤ | eZ α ( X ) | ≤ (cid:18) eC r αn (cid:19) n . Moreover, by the definition of the isotropic constant of µ ,1 = Z K d µ ≤ L nµ | K | . Thus, | A || K | ≤ (cid:18) eCL µ r αn (cid:19) n . We set α such that 4 eCL µ p αn = e − and adjust the constants to finish the proof. (cid:3) As in the proof of Theorem 1, we can assume that µ is isotropic. Let q = q X be definedby (6). Fix 0 < β <
1. By (11) which we apply to the set A = { x ∈ K, q ( x ) > β /κ } ,we have E | K N || K | ≥ | A || K | (cid:18) − (cid:18) Nn (cid:19) (cid:16) − β /κ (cid:17) N − n (cid:19) .
6y the lower bound on q from (17), A ⊃ { x ∈ R n , k x k K ≤ − (16 κ − ) κ β } , hence | A || K | ≥ (cid:0) − (16 κ − ) κ β (cid:1) n ≥ − n (16 κ − ) κ β ≥ − nβ. We choose β such that 32 nβ = ω and crudely deal with the second term, (cid:18) Nn (cid:19) (cid:16) − β /κ (cid:17) N − n ≤ N n e − β /κ ( N − n ) , which is nonincreasing in N as long as N ≥ nβ − /κ . This holds for ω large enough if,say N ≥ n /κ ω /κ . Then we easily conclude that the dominant term above is e − β /κ N which yields, say E | K N || K | ≥ (cid:18) − ω (cid:19) (1 − e − ω n/ ) ≥ − ω , provided that n and ω are large enough. (cid:3) Remark . Groemer’s result used in (5) for uniform distributions has been substantiallygeneralised by Paouris and Pivovarov in [27] to arbitrary distributions with boundeddensities. We remark that in contrast to (5), using the extremality result of the ballfrom [27] does not seem to help obtain bounds from Theorem 2 for two reasons. For one,it concerns bounded densities and rescaling will cost an exponential factor. Moreover,for the example of β -polytopes from [7], we have that they are generated by κ -concavemeasures with κ = β + n and the sharp threshold for the volume is of the order n ( β + n/ (see (3)). The ball would give that N = n (1+ ε ) n/ points is enough. Remark . The example of beta polytopes from (3) shows that the bound on N inTheorem 2 has to be at least of the order n β + n/ = n κ − n/ ≥ n κ . Our bound n κ isperhaps suboptimal. It is not inconceivable that as in the uniform case, the extremalexample is supported on a Euclidean ball. Remark . It is reasonable to ask about sharp thresholds like the ones in (2), (3),(3) and (4) for other sequences of convex bodies, say simplices, cross-polytopes, or ingeneral ℓ p -balls. This is a subject of ongoing work. We refer to [15] for recent resultsestablishing exponential nonsharp thresholds for a simplex (i.e. with a gap between theconstants for lower and upper bounds). References [1] Artstein-Avidan, S., Giannopoulos, A., Milman, V., Asymptotic geometric analy-sis. Part I. Mathematical Surveys and Monographs, 202.
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