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- W1778352006 abstract "Let S be a set of n points in d-space. A convex Steiner partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a convex Steiner partition with at most ⌈(n − 1)/d⌉ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d ≥ 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position.Establishing a tight lower bound for the maximum volume of a tile in a Steiner partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n) in any dimension d ≥ 2. Here we give a (1 − ε)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d-dimensional unit box [0,1] d ." @default.
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- W1778352006 date "2012-01-01" @default.
- W1778352006 modified "2023-09-25" @default.
- W1778352006 title "Minimum Convex Partitions and Maximum Empty Polytopes" @default.
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- W1778352006 doi "https://doi.org/10.1007/978-3-642-31155-0_19" @default.
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