منابع مشابه
Stabbing simplices by points and flats
The following result was proved by Bárány in 1982: For every d ≥ 1 there exists cd > 0 such that for every n-point set S in R d there is a point p ∈ R contained in at least cdn d+1 −O(n) of the d-dimensional simplices spanned by S. We investigate the largest possible value of cd. It was known that cd ≤ 1/(2(d+ 1)!) (this estimate actually holds for every point set S). We construct sets showing ...
متن کاملStabbing Simplices of Point Sets with k-flats
Let S be a set of n points in Rd in general position. A set H of k-flats is called an mk-stabber of S if the relative interior of any m-simplex with vertices in S is intersected by at least one element of H. In this paper we give lower and upper bounds on the size of minimum mk-stabbers of point sets in Rd and in general position. We study mainly mk-stabbers in the plane and in R3.
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We present a deterministic linear time algorithm to find a set of five points that stab a set of n pairwise intersecting disks in the plane. We also give a simple construction with 13 pairwise intersecting disks that cannot be stabbed by three points.
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In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points o...
متن کاملVolume and Lattice Points of Reflexive Simplices
Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected t...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2008
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-008-9124-4