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Keywords:
Lebesgue (signed) measure; polynomial; random vector; real affine variety
Lebesgue (signed) measure; polynomial; random vector; real affine variety
Summary:
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.
References:
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