Some Inequalities in Convex Geometry

Abstract

We present some inequalities in convex geometry falling under the broad theme of quantifying complexity, or deviation from particularly pleasant geometric conditions: we give an upper bound for the Banach--Mazur distance between an origin-symmetric convex body and the $n$-dimensional cube which improves known bounds when n is at least 3 and is "small"; we give the best known upper and lower bounds (for high dimensions) for the maximum number of points needed to hit every member of an intersecting family of positive homothets (or translates) of a convex body, a number which quantifies the complexity of the family's intersections; we give an exact upper bound on the VC-dimension (a measure of combinatorial complexity) of families of positive homothets (or translates) of a convex body in the plane, and show that no such upper bound exists in any higher dimension; finally, we introduce a novel volumetric functional on convex bodies which quantifies deviation from central symmetry, establish the fundamental properties of this functional, and relate it to classical volumetric measures of symmetry.