Borsuk covering and planar sets with unique completion
A Borsuk covering of a bounded set \(S\) in \(\mathbb{R}^ n\) of diameter \(d(S)\) is a covering of \(S\) by a finite number of sets of diameter less than \(d(S)\). The Borsuk number of a set \(S\), denoted by \(b(S)\), is the spallest number of sets forming a Borsuk covering of \(S\). The number \(b(\mathbb{R}^ n)=\{\sup b(S):S\) bounded subset of \(\mathbb{R}^ n\}\) is called the Borsuk number of the space \(\mathbb{R}^ n\). The famous Borsuk problem is whether \(b(\mathbb{R}^ n)=n+1\). It is still open for \(n \geq 4\). In this paper an equivalent formulation of this problem is given. Namely, the following two assertions are equivalent: (i) \(b(\mathbb{R}^ n)=n+1\). (ii) For all convex bodies \(K\) and \(M\) of constant width in \(\mathbb{R}^ n\), we have \(b(K+M)=\{\min b(K),b(M)\}\).
scientific article; zbMATH DE number 500640
scientific article