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.
@InProceedings{aronov_et_al:LIPIcs.SoCG.2016.10, author = {Aronov, Boris and Cheong, Otfried and Dobbins, Michael Gene and Goaoc, Xavier}, title = {{The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {10:1--10:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.10}, URN = {urn:nbn:de:0030-drops-59024}, doi = {10.4230/LIPIcs.SoCG.2016.10}, annote = {Keywords: Union complexity, Convex sets, Motion planning} }
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