The ratios of diameter and width of reduced and of complete convex bodies in Minkowski spaces

Abstract

The breadth of a convex body K within a normed space ${\mathbb {R}}^d$ in direction of a given hyperplane H is the distance between the two supporting hyperplanes of K parallel to H, measured in the underlying norm. The minimal and the maximal value over all directions are called minimal width and diameter of K. The body K is reduced if every body $K_*\subsetneq K$ has a smaller minimal width, and K is complete if every body $K^*\supsetneq K$ has a larger diameter. We study the ratio between diameter and minimal width of reduced and of complete bodies. If $d \ge 3$ then there exist reduced bodies of arbitrarily large ratio, whereas the ratio for complete bodies is bounded by $\frac{d+1}{2}$. As a consequence, every normed space ${\mathbb {R}}^d$, $d \ge 2$, contains reduced bodies that are not complete.