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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)
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)
We study the following local-to-global phenomenon: Let B and R be two finite sets of (blue and red) points in the Euclidean plane R 2. Suppose that in each " neighborhood " of a red point, the number of blue points is at least as large as the number of red points. We show that in this case the total number of blue points is at least one fifth of the total(More)