Solution of the Szökefalvi-Nagy problem for a class of convex polytopes (Q1319972)

\(T(K)\) denote the set of all translates of a convex body \(K\) in \(\mathbb{R}^ d\). The Helly dimension \textit{him} \(T(K)\) of \(T(K)\) is the smallest integer \(m\) such that any finite subfamily \(T'(K) \subseteq T(K)\) has the following property: If every \(m+1\) members of \(T'(K)\) have a common point, then \(\cap T'(K) \neq \emptyset\). The ``Szökefalvi-Nagy problem'' (so-called in papers by Boltyanski and his students) consists in characterizing those convex bodies \(K\) for which \(T(K)\) has a fixed Helly dimension \(r\) \((1 \leq r \leq d)\). The known result by Szökefalvi-Nagy is if \textit{him} \(T(K) = 1\), then \(K\) is a parallelotope. The author gives a complete proof of a result announced in Sov. Math., Dokl. 34, 458-461 (1987); translation from Dokl. Akad. Nauk SSSR 291, 269-272 (1986; Zbl 0626.52009): For zonotopes \(M\) holds \textit{him} \(T(M) \leq r\) if and only if \(M\) is a direct sum of zonotopes, each of dimension no more than \(r\).