On the Euclidean metric entropy of convex bodies

We will explain the subject in a little more detail while briefly describing the organization of the paper. In order to be more precise, let B 2 denote the unit Euclidean ball in R. For a symmetric convex body K ⊂ R let N(K, εB 2 ) be the smallest number of Euclidean balls of radius ε needed to cover K, and finally let M∗(K) be a half of the mean width of K (see (2.1) and (2.2) below). Section 2 collects the notation and preliminary results used throughout the paper. Sudakov’s inequality gives an upper bound for N(K, tB 2 ) in terms of M∗(K), and we show (in Section 3) that if this upper bound is essentially sharp, then diameters of all k-codimensional sections of K are large, for an appropriate choice of k. On the other hand, in Section 4 we discuss conditions that ensure that the covering can be significantly decreased by cutting the body K by a Euclidean ball of a certain radius, in which case “most” of the entropy of K lies outside of this Euclidean ball. In Section 5 this leads to further consequences of sharpness in Sudakov’s inequality which turn out to ∗This author holds the Canada Research Chair in Geometric Analysis.