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

Authors Boris Aronov, Otfried Cheong, Michael Gene Dobbins, Xavier Goaoc

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Boris Aronov
Otfried Cheong
Michael Gene Dobbins
Xavier Goaoc

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Boris Aronov, Otfried Cheong, Michael Gene Dobbins, and Xavier Goaoc. The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.10

Abstract

We show that the union of translates of a convex body in three dimensional space can have a cubic number holes in the worst case, where a hole in a set is a connected component of its compliment. 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.

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Keywords
  • Union complexity
  • Convex sets
  • Motion planning

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