Unique metric segments in the hyperspace over a strictly convex Minkowski space (Q1943374)

Let \((\mathbb{R}^n, \|\cdot\|_B)\) be a normed space with the unit ball \(B\). Let \(\rho _H^B\) be the corresponding Hausdorff metric on the set of compact convex bodies in \(\mathbb{R}^n\). Let \(a, b\in X\), where \((X, \rho)\) is a metric space. A metric segment with endpoints \(a\), \(b\) is a subset of \(X\) isometric to the interval \([0, \rho(a, b)]\). \textit{R.~Schneider} [Bull. Soc. R. Sci. Liège 50, 5--7 (1981; Zbl 0476.52005)] characterized all pairs of convex bodies \(K\) and \(L\), such that they have a unique metric segment joining them, in the case when \(B\) is the Euclidean ball. Later \textit{A. Bogdewicz} and \textit{J. Grzybowski} [Banach Center Publications 84, 75--88 (2009; Zbl 1181.52004)] extended this result to the case of general 2-dimensional convex body. In the paper under consideration, the authors extend Schneider's theorem to the general \(n\)-dimensional case under the assumption that \(B\) is smooth. Namely, they prove that in this case \(K\) and \(L\) have a unique metric segment joining them if and only if either \(K=L+tB\) (or \(L=K+tB\)) for some \(t>0\) or \(K=L+t u\) for some \(t>0\) and \(u\in \partial B\) such that \((T_u B+\ell) \cap L=\{\ell\}\) or \((T_{-u} B+\ell) \cap L=\{\ell\}\) for every \(\ell \in \partial L\), where \(T_a A = \bigcup_{t\geq 0} t(A-a).\)