On maximal primitive fixing systems (Q1922687)

Let \(M\) be a compact convex body in \(\mathbb{R}^n\). A nonzero vector \(v\in \mathbb{R}^n\) is called a moving direction of \(M\) with respect to a subset \(F\subseteq \text{bd} M\) of boundary points of \(M\) if for every \(\lambda >0\) the intersection \(F\cap (v+ \text{int} M)\) is empty. \(F\) is said to be a fixing system of \(M\) (against translations) if there is no moving direction of \(M\) with respect to \(F\). A fixing system \(F\) of \(M\) is primitive if no proper subset \(F'\subset F\) is a fixing system of \(M\). Finally, \(\rho_{\text{max}} (M)\) denote the largest number \(s\) such that there exists a primitive fixing system of \(M\) consisting of \(s\) points. The authors construct three types of convex bodies \(M\) in \(\mathbb{R}^n\) with unbounded \(\rho_{\text{max}} (M)\): a cap body with only one nonregular boundary point, a zonoid with two nonregular boundary points and a zonotop. These nice examples improve results of \textit{B. Bollobás} [Stud. Sci. Math. Hungar. 2, 351-354 (1967; Zbl 0156.43103)] and give new counterexamples to the conjecture \(\rho_{\text{max}} (M) \leq 2(2^n-1)\) of L. Danzer.