Sets in R d with no large empty convex subsets
Discrete Mathematics - Topological, algebraical and combinatorial structures; Froli´k's memorial volume
Discrete & Computational Geometry
An output sensitive algorithm for discrete convex hulls
Computational Geometry: Theory and Applications
Minimum convex partition of a constrained point set
Discrete Applied Mathematics - Special issue 14th European workshop on computational geometry CG'98 Selected papers
Lectures on Discrete Geometry
A Note on Convex Decompositions of a Set of Points in the Plane
Graphs and Combinatorics
Algorithms
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
A fixed parameter algorithm for optimal convex partitions
Journal of Discrete Algorithms
Minimum Weight Convex Steiner Partitions
Algorithmica - Special issue: Algorithms, Combinatorics, & Geometry
Approximation algorithms for the minimum convex partition problem
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
A fixed parameter algorithm for the minimum number convex partition problem
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
Computational geometry column 53
ACM SIGACT News
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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.