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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)
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 H d denote the smallest integer n such that for every convex body K in R d there is a 0 < λ < 1 such that K is covered by n translates of λK. In [2] the following problem was posed: Is there a 0 < λ d < 1 depending on d only with the property that every convex body K in R d is covered by H d translates of λ d K? We prove the affirmative answer to the(More)
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)