Concentration of the Intrinsic Volumes of a Convex Body

Abstract

A. The intrinsic volumes are measures of the content of a convex body. This paper applies probabilistic and information-theoretic methods to study the sequence of intrinsic volumes. The main result states that the intrinsic volume sequence concentrates sharply around a specific index, called the central intrinsic volume. Furthermore, among all convex bodies whose central intrinsic volume is fixed, an appropriately scaled cube has the intrinsic volume sequence with maximum entropy.

Concentration inequalities for ultra log-concave distributions

“…see Corollary 3.1. This improves the bounds 2e −3t 2 /28 from Lotz, McCoy, Nourdin, Peccati, and Tropp [25]. As we will see in Section 3, our bounds give considerable improvement when the central intrinsic volume is of order less than n, since if E[Z K ] is less than, say, √ n, then our bounds improve as the dimension grows to infinity, hence reflecting the high-dimensional aspect of the concentration bound.…”

“…Concentration for the intrinsic volumes of a convex body. In this section, we apply our results to obtain concentration bounds for the intrinsic volumes of a convex body K. Our bounds improve upon [25].…”

“…. , n}, satisfy Var(X) ≤ n, by application of the trivial inequality E[X] ≤ n. Applying this to the particular case where X is the intrinsic volume random variable associated to convex body in R n improves the bound Var(Z K ) ≤ 4n from [25]. Again, since an arbitrary binomial can be realized as the distribution of an intrinsic volume random variable, Proposition 1.3 is sharp for intrinsic volume random variables.…”

“…Using the terminology from [25], the intrinsic volume random variable Z K associated with a convex body K in R n takes non-negative integer values according to the following distribution:…”

“…Many sequences have been proven to be ultra log-concave in graph and matroid theory (see, e.g., [28], [15], [10], [23], [2]). Recently, a fundamental ultra log-concave sequence in convex geometry, namely the intrinsic volumes of a convex body, has been studied in [25], where concentration inequalities are established. Concentration of (conic) intrinsic volumes has been demonstrated to rigorously explain threshold phenomena in recovery of signals and convex optimization problems with random data (see [1]).…”