License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2017.24
URN: urn:nbn:de:0030-drops-72325
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7232/
C. S., Karthik ;
Saha, Arpan
Ham Sandwich is Equivalent to Borsuk-Ulam
Abstract
The Borsuk-Ulam theorem is a fundamental result in algebraic topology, with applications to various areas of Mathematics. A classical application of the Borsuk-Ulam theorem is the Ham Sandwich theorem: The volumes of any n compact sets in R^n can always be simultaneously bisected by an (n-1)-dimensional hyperplane.
In this paper, we demonstrate the equivalence between the Borsuk-Ulam theorem and the Ham Sandwich theorem. The main technical result we show towards establishing the equivalence is the following: For every odd polynomial restricted to the hypersphere f:S^n->R, there exists a compact set A in R^{n+1}, such that for every x in S^n we have f(x)=vol(A cap H^+) - vol(A cap H^-), where H is the oriented hyperplane containing the origin with x as the normal. A noteworthy aspect of the proof of the above result is the use of hyperspherical harmonics.
Finally, using the above result we prove that there exist constants n_0, epsilon_0>0 such that for every n>= n_0 and epsilon <= epsilon_0/sqrt{48n}, any query algorithm to find an epsilon-bisecting (n-1)-dimensional hyperplane of n compact set in [-n^4.51,n^4.51]^n, even with success probability 2^-Omega(n), requires 2^Omega(n) queries.
BibTeX - Entry
@InProceedings{cs_et_al:LIPIcs:2017:7232,
author = {Karthik C. S. and Arpan Saha},
title = {{Ham Sandwich is Equivalent to Borsuk-Ulam}},
booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)},
pages = {24:1--24:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-038-5},
ISSN = {1868-8969},
year = {2017},
volume = {77},
editor = {Boris Aronov and Matthew J. Katz},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7232},
URN = {urn:nbn:de:0030-drops-72325},
doi = {10.4230/LIPIcs.SoCG.2017.24},
annote = {Keywords: Ham Sandwich theorem, Borsuk-Ulam theorem, Query Complexity, Hyperspherical Harmonics}
}
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Keywords: |
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Ham Sandwich theorem, Borsuk-Ulam theorem, Query Complexity, Hyperspherical Harmonics |
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Collection: |
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33rd International Symposium on Computational Geometry (SoCG 2017) |
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Issue Date: |
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2017 |
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Date of publication: |
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20.06.2017 |
