Maximal primitive hindering systems for convex figures (Q703688)

The notion of a hindering system for convex bodies was introduced by P. Mani (1971), and studied later by the authors of the present paper (2001 and 2003). Let \(M \subset\mathbb R^n\) be a compact, convex body and \(F \subset\) bd\(M\). The set \(F\) is a hindering system for \(M\) if for every nonzero vector \(v\) there exists a \(\lambda >0\) such that \((M+\lambda v) \cap F \neq \emptyset.\) A hindering system for \(M\) is primitive if none of its proper subsets is a hindering system for \(M\). Let \(\sigma_{\max}M\) be the maximal cardinality of primitive hindering systems for \(M\). P. Mani obtained some results concerning \(\sigma_{\max}M\) for \(M \subset {\mathbb R}^2\), in particular he obtained the inequality \(3 \leq \sigma_{\max}M \leq 5\). In the present paper the authors give a complete classification of compact convex planar figures with respect to \(\sigma_{\max}\) (Theorems 1 and 2). Further, Theorem 3 concerns figures of constant width: \(\sigma_{\max}M\) is equal to 3 if \(M\) is the Reuleaux triangle, is equal to 5 if \(M\) is a Reuleaux pentagon, and is equal to 4 otherwise. Theorem 4 concerns regular polygons: if \(M_k\) is a regular polygon with \(k\) vertices, then \(\sigma_{\max}\) is equal to \(k\) for \(k \leq 5\) and is equal to 4 for \(k>5\). Proofs and examples are illustrated by 15 figures.