Peter Pivovarov

Small-Ball Probabilities for the Volume of Random Convex Sets

We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.

On the volume of caps and bounding the mean-width of an isotropic convex body

Let K be a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on K exhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width.

On sharp bounds for marginal densities of product measures

We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if f is a probability density on Rn of the form f(x) = Пi=1nfi(xi), where each fi is a density on R, say bounded by one, then the density of any marginal πE(f) is bounded by 2k/2, where k is the dimension of E. The proof relies on an adaptation of Ball’s approach to cube slicing, carried out for functions. Motivated by inequalities for dual affine quermassintegrals, we also prove an isoperimetric inequality for certain averages of the marginals of such f for which the cube is the extremal case.

Random ball-polyhedra and inequalities for intrinsic volumes

We prove a randomized version of the generalized Urysohn inequality relating mean width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the centers have i.i.d. coordinates, then the uniform measure on a cube is the extremizer. We also discuss connections to a randomized version of the extended isoperimetric inequality and symmetrization techniques.

Intrinsic volumes and linear contractions

It is shown that intrinsic volumes of a convex body decrease under linear contractions. Let C ⊂ R be a convex body and B 2 the Euclidean ball in R . The Steiner formula expresses the volume of the Minkowski sum C + eB 2 in terms of the intrinsic volumes V0, V1, . . . , VN of C: volN ( C + eB 2 ) = N ∑ n=0 ωnVN−n(C)e . Here volN (·) denotes N -dimensional Lebesgue measure and ωn = voln (B 2 ). Of particular interest are V1, VN−1 and VN , which are multiples of the mean-width, surface area and volume, respectively. We refer the reader to [5] for background on intrinsic volumes. In addition to their role in convex geometry, intrinsic volumes also appear in connection with Gaussian processes; see, e.g., [9], [10] and the references therein. The purpose of this note is to prove the following. Proposition 1.1. Let C ⊂ R be a convex body and let S be a linear contraction, i.e., ‖Sx‖2 6 ‖x‖2 for each x ∈ R . Then for n = 1, . . . , N , Vn(SC) 6 Vn(C). The case of V1 and arbitrary contractions (not necessarily linear) is well-studied [6, Theorem 2 in §5], [1, Theorem 1]; see also [2, pg 177]. Of course for VN one has VN(SC) = |det(S)| volN (C). For other intrinsic volumes, we were unable to find Proposition 1.1 in the literature but noticed that it follows from some results in [4] and thought it was worthwhile to show the details. ∗The first-named author is supported by the US National Science Foundation, grant DMS-0906150. †The second-named author holds a Postdoctoral Fellowship award from the Natural Sciences and Engineering Research Council of Canada.

Randomized Isoperimetric Inequalities

We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities admit stronger randomized forms in the following sense: for natural families of associated random convex sets one has stochastic dominance for various functionals such as volume, surface area, mean width and others. By laws of large numbers, these randomized versions recover the classical inequalities. We give an overview of when such stochastic dominance arises and its applications in convex geometry and probability.

Random Convex Bodies Lacking Symmetric Projections, Revisited Through Decoupling

In 2001, E.D. Gluskin, A.E. Litvak and N. Tomczak-Jaegermann, using probabilistic methods inspired by some earlier work of Gluskin’s, provided an example of a convex body K R n such that no suitably large rank projection of K is symmetric. We provide an alternate proof of the existence of such a body, the key ingredient of which is a decoupling result due to S.J. Szarek and Tomczak-Jaegermann. In problems that seem susceptible to probabilistic methods, independence is often desirable but not necessarily present within the given constraints. A recent decoupling result due to S.J. Szarek and N. Tomczak-Jaegermann allows one to overcome the obstacle of dependency, given that certain conditions are present. Originally applied to some problems in the asymptotic theory of normed spaces [13], the general statement in an arbitrary probabilistic setting appears in [12]. This note presents a natural application of said decoupling result in the theory of non-symmetric convex bodies. Namely, we show how it can be used to provide a new proof of a theorem due to E.D. Gluskin, A.E. Litvak and Tomczak-Jaegermann [6]. The theorem asserts there is a convex body K R n such that for any projection P, the Minkowski measure of symmetry of PK is at least (rankP)/c p nlnn, where c is a positive absolute constant. Besides using Gaussian random vectors instead of uniformly distributed random vectors on the sphere, we follow the same approach as the The author holds a Natural Sciences and Engineering Research Council of Canada Post