Reduced Convex Bodies in Finite Dimensional Normed Spaces: A Survey

Abstract

For a convex body C in a finite dimensional real Banach space M ^d denote by ${\triangle(C)}$ its thickness, i.e., its minimal width with respect to the norm. A convex body ${R \subset M^d}$ is said to be reduced if ${\triangle(C) < \triangle(R)}$ for each convex body C properly contained in R. The concept of reduced bodies is particularly useful for solving volume-minimizing problems in the area of convexity, and it is also important as extension of basic notions from convexity and functional analysis. Namely, on the one hand the class of reduced bodies is a “dualization” of the concept of complete sets (and, in Euclidean space, that of constant width). On the other hand, it forms a proper superset of the class of complete sets. We present the recent knowledge on this class of convex bodies in finite dimensional real Banach spaces. First we collect general properties of arbitrary reduced bodies. For example, we present constructions supporting our conjecture that in any normed space of dimension larger than 2 there are reduced bodies of unit thickness and diameter at least λ, for every positive number λ. Then we will lay special emphasize on reduced polytopes, and finally on the geometric description of planar reduced bodies. The survey also presents several research problems.