On generalized ``Ham Sandwich'' theorems

Marek Golasinski

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 87-100 Torun, Chopina 12/18, Poland

E-mail. marek@mat.uni.torun.pl

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'': {\em 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 $\{x\in \mathbb{R}^n;\,\lambda_0f_0(x)+\cdots +\lambda_mf_m(x)=0 \}$ simultaneously bisects each of subsets $A_k$, $k=1,\ldots,m$.} Then some its applications are studied.

AMSclassification. Primary 58C07; Secondary 12D10, 14P05.

Keywords. Lebesgue (signed) measure, polynomial, random vector, real affine variety.