Fixing and hindering systems of planar convex figures. (Q1432616)

Recall that a subset \(F\) of the boundary of a planar convex body \(M\) is called a fixing system for \(M\) if for every vector \(\vec v \not = \vec 0\) there exists a number \(\lambda >0\) such that \((\lambda \vec v + \text{ int} M) \cap F \not = \emptyset\). We call \(F\) a hindering system for \(M\) if for every vector \(\vec v \not = \vec 0\) there exists a number \(\lambda >0\) such that \((\lambda \vec v + M) \cap F \not = \emptyset\). We say that a fixing (hindering) system of \(M\) is primitive if it is not a proper subset of a fixing (respectively: a hindering) system of \(M\). Denote by \(\rho_{\max}M\) the greatest cardinality of a primitive fixing system of \(M\) and by \(\sigma_{\max}M\) the greatest cardinality of a primitive hindering system of \(M\). \textit{L. Fejes Tóth} [Acta Math. Acad. Sci. Hung. 13, 379--382 (1962; Zbl 0113.16203)] proved that \(3 \leq \rho_{\max}M \leq 6\) and \textit{P. Mani} [Acta Math. Acad. Sci. Hung. 22, 269--273 (1972; Zbl 0251.52010)] proved that \(3 \leq \sigma_{\max}M \leq 5\) for every planar convex body \(M\). Then other authors characterized convex bodies \(M\) with \(\rho_{\max}M = 6\). The authors of this paper present some theorems (without proofs) which characterize convex bodies fulfilling the equalities \(\rho_{\max}M = 3\), \(\rho_{\max}M = 5\), \(\sigma_{\max}M = 3\) and \(\sigma_{\max}M = 5\). So we get characterizations of \(M\) with \(\rho_{\max}M = i\) for every possible \(i\) and of \(M\) with \(\sigma _{\max}M = j\) for every possible \(j\).