On generalized “ham sandwich” theorems

Marek Golasiński

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 1, page 25-30
  • ISSN: 0044-8753

Abstract

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In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let A 1 , ... , A m ⊆ ℝ n be subsets with finite Lebesgue measure. Then, for any sequence f 0 , ... , f m of ℝ -linearly independent polynomials in the polynomial ring ℝ [ X 1 , ... , X n ] there are real numbers λ 0 , ... , λ m , not all zero, such that the real affine variety { x ∈ ℝ n ; λ 0 f 0 ( x ) + ⋯ + λ m f m ( x ) = 0 } simultaneously bisects each of subsets A k , k = 1 , ... , m . Then some its applications are studied.

How to cite

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Golasiński, Marek. "On generalized “ham sandwich” theorems." Archivum Mathematicum 042.1 (2006): 25-30. <http://eudml.org/doc/249826>.

@article{Golasiński2006,
abstract = {In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let $A_1,\ldots ,A_m\subseteq \mathbb \{R\}^n$ be subsets with finite Lebesgue measure. Then, for any sequence $f_0,\ldots ,f_m$ of $\mathbb \{R\}$-linearly independent polynomials in the polynomial ring $\mathbb \{R\}[X_1,\ldots ,X_n]$ there are real numbers $\lambda _0,\ldots ,\lambda _m$, not all zero, such that the real affine variety $\lbrace x\in \mathbb \{R\}^n;\,\lambda _0f_0(x)+\cdots +\lambda _mf_m(x)=0 \rbrace $ simultaneously bisects each of subsets $A_k$, $k=1,\ldots ,m$. Then some its applications are studied.},
author = {Golasiński, Marek},
journal = {Archivum Mathematicum},
keywords = {Lebesgue (signed) measure; polynomial; random vector; real affine variety; Lebesgue (signed) measure; polynomial; random vector; real affine variety},
language = {eng},
number = {1},
pages = {25-30},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On generalized “ham sandwich” theorems},
url = {http://eudml.org/doc/249826},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Golasiński, Marek
TI - On generalized “ham sandwich” theorems
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 1
SP - 25
EP - 30
AB - In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let $A_1,\ldots ,A_m\subseteq \mathbb {R}^n$ be subsets with finite Lebesgue measure. Then, for any sequence $f_0,\ldots ,f_m$ of $\mathbb {R}$-linearly independent polynomials in the polynomial ring $\mathbb {R}[X_1,\ldots ,X_n]$ there are real numbers $\lambda _0,\ldots ,\lambda _m$, not all zero, such that the real affine variety $\lbrace x\in \mathbb {R}^n;\,\lambda _0f_0(x)+\cdots +\lambda _mf_m(x)=0 \rbrace $ simultaneously bisects each of subsets $A_k$, $k=1,\ldots ,m$. Then some its applications are studied.
LA - eng
KW - Lebesgue (signed) measure; polynomial; random vector; real affine variety; Lebesgue (signed) measure; polynomial; random vector; real affine variety
UR - http://eudml.org/doc/249826
ER -

References

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  9. Peters J. V., The ham sandwich theorem for some related results, Rocky Mountain J. Math. 11 (3) (1981), 473–482. (1981) MR0722580
  10. Steinhaus H., Sur la division des ensembles de l’espace par les plans et des ensembles plans par les cercles, Fund. Math. 33 (1945), 245–263. (1945) Zbl0061.38404MR0017514
  11. Steinhaus H., Kalejdoskop matematyczny, PWN, Warszawa (1956). (1956) 
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