Random aspects of high-dimensional convex bodies

Abstract

In this paper we study geometry of compact, not necessarily centrally symmetric, convex bodies in R. Over the years, local theory of Banach spaces developed many sophisticated methods to study centrally symmetric convex bodies; and already some time ago it became clear that many results, if valid for arbitrary convex bodies, may be of interest in other areas of mathematics. In recent years many results on non-centrally symmetric convex bodies were proved and a number of papers have been written (see e.g., [1], [8], [12], [18], [27], [28] among others). The present paper concentrates on random aspects of compact convex bodies and investigates some invariants fundamental in the local theory of Banach spaces, restricted to random sections and projections of such bodies. It turns out that, loosely speaking, such random operations kill the effect of non-symmetry in the sense that resulting estimates are very close to their centrally symmetric counterparts (this is despite the fact that random sections might be still far from being symmetric (see Section 5 below)). At the same time these estimates might be in a very essential way better than for general bodies. We are mostly interested in two directions. One is connected with so-called MM∗estimate, and related inequalities. For a centrally symmetric convex body K ⊂ R, an estimate M(K)M(K) ≤ c log n (see the definitions in Section 2 below) is an important technical tool intimately related to the Kconvexity constant. It follows by combining works by Lewis and by Figiel and Tomczak-Jaegermann, with deep results of Pisier on Rademacher projections (see e.g., [26]). Although the symmetry can be easily removed from the first two parts, Pisier’s argument use it in a very essential way. In Section 4 we show, in particular, that every convex body K has a position K1 (i.e., K1 = uK − a for some operator u and a ∈ R) such that a random projection, PK1, of dimension [n/2] satisfies M(PK1)M(K 1 ) ≤ C log n, where C is an absolute constant. Moreover, there exists a unitary operator u such that M(K1 +uK1)M(K 1 ) ≤ C log n. Our proof is based essentially on symmetric considerations, a non-symmetric part is reduced to classical facts and simple