Functions with comparable integrals on all k-planes

Abstract

Let $d\ge 2$ and $1\le k\le d-1$. We show that if $f:{\mathbb{R}}^d\to \mathbb{R}$ is nonnegative and measurable and $0<m\le M<\infty$ then it is impossible that on almost all affine $k$-planes $P$ in ${\mathbb{R}}^d$ the integral of $f$ on $P$ lies between $m$ and $M$. Let $G={\mathbb{Z}}^k\times \{0{\}}^{d-k}$. Using $m=M=1$ and $f$ being the indicator function of a measurable set in ${\mathbb{R}}^d$ this then implies that there is no measurable Steinhaus set for the group $G$ in ${\mathbb{R}}^d$. In other words there is no measurable set $S\subseteq {\mathbb{R}}^d$ such that $S$ tiles ${\mathbb{R}}^d$ with $T(G)$, for all $T\in O(d)$. We also generalize our impossibility results concerning the size of line-integrals of functions to integrals on strips in the plane.

8 pages