On generalized “ham sandwich” theorems
Golasiński, Marek
Archivum Mathematicum, Tome 042 (2006), p. 25-30 / Harvested from Czech Digital Mathematics Library
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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 $A_1,\ldots ,A_m\subseteq \mathbb {R}^n$ 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.

Publié le : 2006-01-01
Classification:  12D10,  14P05,  58C07 
@article{107978,
     author = {Marek Golasi\'nski},
     title = {On generalized ``ham sandwich'' theorems},
     journal = {Archivum Mathematicum},
     volume = {042},
     year = {2006},
     pages = {25-30},
     zbl = {1164.58312},
     mrnumber = {2227109},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107978}
}

Golasiński, Marek. On generalized “ham sandwich” theorems. Archivum Mathematicum, Tome 042 (2006) pp. 25-30. http://gdmltest.u-ga.fr/item/107978/

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