Successive radii and Minkowski addition (Q431178)

Given a convex body \(K \subset {\mathbb R}^n \), the authors formulate definitions of its successive outer radii \( R_i (K) \) and inner radii \( r_i (K) \): For \( i = 1, \ldots n \) let \( R_i (K) = \min_{L \in {\mathcal L}_i^n} R(K \mid L) \), and \[ r_i (K) = \max_{ L \in {\mathcal L}_i^n} \left( \max_{x \in L^{\bot}} r(K \cap (x+L); x+L) \right) . \] Here \( {\mathcal L}_i^n \) is the set of all \(i\)-dimensional subspaces of \({\mathbb R}^n \), \( R(K)\) and \( r(K) \) denote respectively the circumradius and the inradius of \(K\), and \( K \mid L \) is the orthogonal projection of \( K \) onto \( L\). One of the main results of the paper is Theorem 1.1. Let \( K, K'\) be convex bodies in \( {\mathbb R}^n \). Then \[ \begin{aligned} R_1 (K + K') \geq & R_1 (K) + R_1 (K'), \\ \sqrt{2} R_i (K + K') \geq & R_i (K) + R_i (K'); \quad i = 2, \dots n. \end{aligned} \] All inequalities are best possible. Similar inequalities are proved for the inner successive radii. Some special cases, when \( K'\) is the Euclidean ball, and the case of \( K + (-K) \) are considered as well.