Relative isodiametric inequalities (Q1878964)

Let \(G\) be a bounded open convex set in \(\mathbb{R}^{n}\) and \(E\) a subset of \(G\) such that \(E\) as well as \(G\backslash E\) are connected and have non-empty interior. The authors study inequalities concerning the ``relative volume'' \(V(E,G)=\min \{V(E),V(G\backslash E)\}\), the ``minimum relative diameter'' \(d_{m}(E,G)=\min \{D(E),D(G\backslash E)\}\), and the ``maximum relative diameter'' \(d_{M}(E,G)=\max \{D(E),D(G\backslash E)\}\). Here \(V(E)\) and \(D(E)\) denote volume and diameter of \(E\), respectively. It is shown that for each \(G\) the number \(0\) is the best possible lower bound for the quotients \(V(E,G)/d_{m}(E,G)^{n}\) and \(V(E,G)/d_{M}(E,G)^{n}\), even if \(E\) is restricted to sets obtained by a hyperplane cut. The first quotient is bounded above by the ratio \(V(B)/D(B)^{n}\) of a ball \(B\). A best upper bound for the second quotient is obtained only for planar sets in case \(E\) is obtained by a straight line cut. The numerical value of this bound is \( 1.2869\ldots \). It is attained for a certain set \(G\) bounded by four circular arcs and two straight line segments, which has been already studied in a previous paper [\textit{C. Miori, C. Peri}, and \textit{S. Segura Gomis}, J. Math. Anal. Appl. 300, No. 2, 464--476 (2004; Zbl 1080.52002)].