On the Borsuk and Grünbaum problems for \((0,1)\)- and \((-1,0,1)\)-polytopes in low-dimensional spaces (Q352177)

This paper is a summary of the results in [\textit{V. B. Goldshteyn}, Tr. Mosk. Fiz. Tekh. Inst. 4, No. 1, 91--110 (2012)]. It concerns two problems related to the diameter of a bounded subset \(V \subseteq \mathbb{R}^n\), where \(\text{diam}(V)\) is defined as \(\sup\{ \rho(x,y): x,y \in V\}\) and \(\rho\) denotes the standard Euclidean distance. Let \(V \subseteq \mathbb{R}^n\) be a bounded set. Borsuk's problem concerns the minimal number of parts in a partition of \(V\) such that each part has diameter strictly smaller than that of \(V\). The function \(f(V)\) denotes the minimal number of such subsets required to cover a given set \(V\); and \(f(n)\) is the maximal value of \(f(V)\) over all bounded subsets \(V \subseteq \mathbb{R}^n\). Grünbaum's problem concerns the problem of partitioning \(V\) into sets that can be enclosed by a ball whose diameter equals that of \(V\). The functions \(g(V)\) and \(\overline{g}(V)\) respectively denote the minimal number of open (respectively closed) balls of diameter \(\text{diam}(V)\) required to cover \(V\); and \(g(n)\) (respectively \(\overline{g}(n)\)) denotes the maximal values of \(g(V)\) (respectively \(\overline{g}(V)\)) as \(V\) ranges over all bounded subsets of \(\mathbb{R}^n\). Borsuk proved that \(f(B^n) = n+1\) (\(B^n\) denotes an \(n\)-ball), and asked if \(f(n) = n+1\) in general. As a generalization of this, Grünbaum conjectured that \(\overline{g}(n) = n+1\). Counterexamples have been given to both of these questions in general when \(n\) is sufficiently large (see \textit{J. Kahn} and \textit{G. Kalai} [Bull. Am. Math. Soc., New Ser. 29, No. 1, 60--62 (1993; Zbl 0786.52002)] and \textit{J. Bourgain} and \textit{J. Lindenstrauss} [Lect. Notes Math. 1469, 138--144 (1991; Zbl 0815.46017)]). The results summarized in this paper show that when \(n\) is small and \(V\) is a subset of \(\{0,1\}^n\) or \(\{-1,0,1\}^n\), both Borsuk's conjecture and Grünbaum's conjecture hold.