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In this paper we introduce ball-polyhedra as finite intersections of congruent balls in Euclidean 3-space. We define their duals and study their face-lattices. Our main result is an analogue of Cauchy's rigidity theorem. Take the intersection of finitely many but at least three closed balls of given radius, say R > 0, in Euclidean 3-space and assume that… (More)

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a… (More)

Let F be a family of positive homothets (or translates) of a given convex body K in R n. We investigate two approaches to measuring the complexity of F. First, we find an upper bound on the transversal number τ (F) of F in terms of n and the independence number ν(F). This question is motivated by a problem of Grünbaum [2]. Our bound τ (F) ≤ 2 n 2n n (n log… (More)

Lovász and Stein (independently) proved that any hypergraph satisfies τ ≤ (1+ln ∆)τ * , where τ is the transversal number, τ * is its fractional version, and ∆ denotes the maximum degree. We prove τ f ≤ 3.17τ * max{ln ∆, f } for the f-fold transversal number τ f. Similarly to Lovász and Stein, we also show that this bound can be achieved… (More)

—A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. A new result of this paper is a Cauchy-type rigidity theorem for ball-polyhedra. Its proof presented here is based on the underlying truncated Voronoi and Delaunay complexes of ball-polyhedra.

We introduce a fractional version of the illumination problem of Gohberg, Markus, Boltyanski and Hadwiger, according to which every convex body in R d is illuminated by at most 2 d directions. We say that a weighted set of points on S d−1 illuminates a convex body K if for each boundary point of K, the total weight of those directions that illuminate K at… (More)

We say that a convex set K in R d strictly separates the set A from the set B if A ⊂ int(K) and B ∩ cl K = ∅. The well–known Theorem of Kirchberger states the following. If A and B are finite sets in R d with the property that for every T ⊂ A ∪ B of cardinality at most d + 2, there is a half space strictly separating T ∩ A and T ∩ B, then there is a half… (More)