Abstract
The aim of this text is to provide an elementary and self-contained exposition of Gromov’s argument on topological overlap (the presentation is based on Gromov’s work, as well as two follow-up papers of Matoušek and Wagner, and of Dotterrer, Kaufman and Wagner). We also discuss a simple generalization in which the vertices are weighted according to some probability distribution. This allows to use von Neumann’s minimax theorem to deduce a dual statement.
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Notes
Some triangles intersect in an edge, and a small region close to the boundary of B is not covered.
The symbol \(\pitchfork \) seems to be a drawing of an intersection.
The notation does not indicate if \(\delta \) acts on sets of vertices or edges, but this is clear from the context.
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Acknowledgements
I thank Shay Moran for helpful suggestions and comments. I also thank the anonymous referees for very helpful suggestions. Research was supported by ISF Grant 1162/15.
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Yehudayoff, A. An Elementary Exposition of Topological Overlap in the Plane. Discrete Comput Geom 58, 255–264 (2017). https://doi.org/10.1007/s00454-017-9910-y
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DOI: https://doi.org/10.1007/s00454-017-9910-y