Combinatorial geometry of belt bodies
Belt polytopes are natural generalizations of zonotopes. A polytope \(P \subset \mathbb{R}^d\) is a belt polytope, if in every 2-face of \(P\) the sides occur in parallel pairs. If, in addition, the parallel sides have the same length, then \(P\) is a zonotope. The authors introduce a new class of convex bodies, the belt bodies. For that purpose, a unit vector \(u\) is called a belt vector of a convex body \(K\), if every supporting line \(L\) parallel to \(u\) meets \(K\) at a unique point \(a\) and if one of the two rays with direction \(\pm u\) starting in \(a\) is a tangential ray at \(K\). Let \(B(K)\) be the set of all belt vectors of \(K\) and \(p(B(K))\) the unit vectors orthogonal to linearly independent \((d-1)\)-tuples from \(B(K)\). \(K\) is called a belt body if \(p(B(K))\) is dense in the set of outer unit normals of \(K\) at regular boundary points. The authors show that zonoids are belt bodies and that a polytope is a belt body if and only if it is a belt polytope. Consequently, belt bodies are dense in the class of all convex bodies. The main purpose of the work is to solve three well-known (and still unsolved) combinatorial problems for the class of belt bodies. The first problem is to cover a convex body \(K\) with a minimal number \(b(K)\) of translated smaller homothets of \(K\) (a problem due to Hadwiger, Gohberg and Markus). The covering problem has an equivalent formulation in terms of illumination of \(K\). For belt bodies \(K\) (which are not parallelotopes), the inequality \[ b(M) \leq 3 \cdot 2^{d-2} \] is shown. The second problem is to find the Helly dimension of the family of all translates of a given convex body \(K\) (Szökefalvi-Nagy problem). Here, the authors give a complete description of the Helly dimension of belt bodies. Finally, the latter result is applied to minimal fixing systems of convex bodies (a notion which goes back to Fejes Tóth). Again, for belt bodies \(K\), a complete description of the minimal fixing number is given. Some of the results are also contained in \textit{V. Boltyanski} [Mat. Zametki (to appear)] and \textit{E. Baladze} and \textit{V. Boltyanski} [Mat. Sbornik (to appear)].
scientific article; zbMATH DE number 828274
scientific article