On the volume of sets having constant width (Q1110813)

Let C be a set of constant width d in \({\mathbb{R}}^ n\). Besides other interesting results, a lower bound for the volume of C is given. Namely, if Vol B(d/2) denotes the n-dimensional volume of a ball B in \({\mathbb{R}}^ n\) with radius d/2, then \[ Vol C\geq (\sqrt{3+2/(n+1)}-1)^ n\cdot Vol B(d/2) \] holds. For \(n>4\), this is an improvement of well-known results obtained by \textit{W. Firey} [Arch. Math. 16, 69-74 (1965; Zbl 0128.164)] and \textit{G. D. Chakerian} [Pac. J. Math. 19, 13-21 (1966; Zbl 0142.207)].