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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. © 2004 Elsevier Ltd. All rights reserved.
Let Hd denote the smallest integer n such that for every convex body K in R there is a 0 < λ < 1 such that K is covered by n translates of λK. In  the following problem was posed: Is there a 0 < λd < 1 depending on d only with the property that every convex body K in R is covered by Hd translates of λdK? We prove the affirmative answer to the question… (More)
Bárány, Katchalski and Pach (Proc Am Math Soc 86(1):109–114, 1982) (see also Bárány et al., Am Math Mon 91(6):362–365, 1984) proved the following quantitative form of Helly’s theorem. If the intersection of a family of convex sets in R d is of volume one, then the intersection of some subfamily of at most 2d members is of volume at most some constant v(d).… (More)
Let F be a family of positive homothets (or translates) of a given convex body K in Rn. 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 . Our bound τ(F) ≤ 2n (2n n ) (n… (More)
We prove that for every convex body K with the center of mass at the origin and every ε ∈ ( 0, 12 ) , there exists a convex polytope P with at most eO(d)ε− d−1 2 vertices such that (1− ε)K ⊂ P ⊂ K.
A ball-polyhedron is the intersection with nonempty 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. Keywords-ball-polyhedron; dual… (More)
We say that a convex set K in R strictly separates the set A from the set B if A ⊂ int(K) and B ∩ clK = ∅. The well–known Theorem of Kirchberger states the following. If A and B are finite sets in R 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 space… (More)