The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions

We show that the union of n translates of a convex body in  R 3 can have  Θ ( n 3 ) holes in the worst case, where a hole in a set  X is a connected component of R 3 \ X . This refutes a 20-year-old conjecture. As a consequence, we also obtain improved lower bounds on the complexity of motion planni...

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Bibliographic Details
Published inDiscrete & computational geometry Vol. 57; no. 1; pp. 104 - 124
Main Authors Aronov, Boris, Cheong, Otfried, Dobbins, Michael Gene, Goaoc, Xavier
Format Journal Article
LanguageEnglish
Published New York Springer US 2017
Springer Nature B.V
Springer Verlag
Subjects
Combinatorics
Complexity
Computational Geometry
Computational Mathematics and Numerical Analysis
Computer Science
Geometry
Lower bounds
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Motion planning
Texts
Theoretical mathematics
Unions
Motion planning
Convex sets
Union complexity
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Summary:We show that the union of n translates of a convex body in  R 3 can have  Θ ( n 3 ) holes in the worst case, where a hole in a set  X is a connected component of R 3 \ X . This refutes a 20-year-old conjecture. As a consequence, we also obtain improved lower bounds on the complexity of motion planning problems and of Voronoi diagrams with convex distance functions.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 23
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-016-9820-4