Chapter 4 Convex geometry and functional analysis

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This chapter describes three topics that lie at the intersection of functional analysis, harmonic analysis, probability theory, and convex geometry. The classical isoperimetric inequality states that among the bodies of a given volume in R n , the Euclidean balls have least surface area. This principle seems to have been recognized, at least in two dimensions, by the ancients and by the end of the past century, there were a number of proofs that worked in arbitrary dimension. The simplex has the largest volume ratio among the convex bodies of a given dimension, while among the symmetric bodies the cube is extremal. The Loomis–Whitney inequality already looks a bit like an isoperimetric inequality, because it estimates the volume of K in terms of the volumes of 1-codimensional sets derived from K . In fact, a simple generalization of the Loomis–Whitney inequality is the main technical tool in Gagliardo's proof of the Sobolev embedding theorem. The simplex has the largest volume ratio of any convex body and that the cube has the largest of any symmetric body. For a convex body whose maximal ellipsoid is known, these volume ratio estimates automatically provide upper bounds for the volume of the body.