On extensive subsets of convex bodies (Q5932662)

Let \(K\) be a \(d\)-dimensional convex body. The author calls a subset \(S\) of its boundary extensive if the line segment from \(p\) to \(q\) has at least half the maximal length of a line segment parallel to \(pq\) that is contained in \(K\). It is proved that an extensive subset of \(K\) has size at most \(3^d-1\), thus extending a result of \textit{H. Hadwiger} [Arch. Math. 8, 212-213 (1957; Zbl 0080.15501)] for the case of difference bodies \(K\) and \(-K\), which bounds translative kissing numbers. Extensive subsets of special \(3\)-dimensional polytopes, in particular of octahedra, are then studied in detail.