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- Károly Bezdek, Márton Naszódi
- Eur. J. Comb.
- 2006

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)

- Károly Bezdek, Márton Naszódi
- Discrete & Computational Geometry
- 2013

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)

- Márton Naszódi, Steven Taschuk
- Discrete Mathematics
- 2010

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)

- Balázs Csikós, Zsolt Lángi, Márton Naszódi
- Periodica Mathematica Hungarica
- 2006

- Márton Naszódi
- Discrete & Computational Geometry
- 2016

- Márton Naszódi
- Contributions to Discrete Mathematics
- 2009

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)

- Márton Naszódi, Alexandr Polyanskii
- ArXiv
- 2016

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)

- Zsolt Lángi, Márton Naszódi, János Pach, Gábor Tardos, Géza Tóth
- J. Comb. Theory, Ser. A
- 2016

- Zsolt Lángi, Márton Naszódi
- Periodica Mathematica Hungarica
- 2008

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)