## On generalized ``Ham Sandwich'' theorems

##
*Marek Golasinski*

**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.