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.