On the volume of equichordal sets

Abstract

IfK is an equichordal set of chord length 1, i.e. ann-dimensional convex body with a pointp εK such that every chord throughp has length 1, it can be shown that ω _n /2ⁿ ⩽v(K) < ω _n /2, wherev(K) denotes the volume ofK and ω _n the volume of ann-dimensional unit ball. Explicit estimates are established for the deviation ofK from a ball of radius 1/2 ifv(K)−ω _n /2ⁿ is small, and from a semiball of radius 1 if 1/2ω _n −v(K) is small.