Address.
Faculty of Mathematics and Computer Science,
Nicolaus Copernicus University,
87-100 Torun, Chopina 12/18, Poland
E-mail. marek@mat.uni.torun.pl
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'':
{\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.