On the volume of equichordal sets. (Q1107825)

If \(K\) is an equichordal set of chord length 1, i.e. an \(n\)-dimensional convex body with a point \(p\in K\) such that every chord through \(p\) has length 1, it can be shown that \(\omega_1/2^n\leq v(K)<\omega_n/2\), where \(v(K)\) denotes the volume of \(K\) and \(\omega_n\) the volume of an \(n\)-dimensional unit ball. Explicit estimates are established for the deviation of \(K\) from a ball of radius \(1/2\) if \(v(K)-\omega_n/2^n\) is small, and from a semiball of radius 1 if \(\frac12\omega_n-v(K)\) is small.