On the volume of equichordal sets

IfK is an equichordal set of chord length 1, i.e. ann-dimensional convex body with a pointp K such that every chord throughp has length 1, it can be shown that _n /2 ⁿ v(K) < _n /2, wherev(K) denotes the volume ofK and _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)– _n /2 ⁿ is small, and from a semiball of radius 1 if 1/2 _n –v(K) is small.Supported by National Science Foundation Research Grant DMS 8701893.