Márton Naszódi

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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 [2]. Our bound τ(F) ≤ 2n (2n n ) (n(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)