 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://drops.dagstuhl.de/opus/volltexte/2017/7232/
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### Ham Sandwich is Equivalent to Borsuk-Ulam

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### 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}
}
```

 Keywords: Ham Sandwich theorem, Borsuk-Ulam theorem, Query Complexity, Hyperspherical Harmonics Seminar: 33rd International Symposium on Computational Geometry (SoCG 2017) Issue Date: 2017 Date of publication: 08.06.2017

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