在所有k-平面上具有可比积分的函数
Functions with comparable integrals on all k-planes
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中文总结 AI 辅助
该研究针对d≥2、1≤k≤d-1的情况，证明非负可测函数无法在几乎所有仿射k-平面上的积分都介于两个正数之间，进而推出不存在相关可测Steinhaus集，并将线积分的不可能性结果推广到平面带形积分。
中文摘要 AI 辅助
设d≥2且1≤k≤d-1，我们证明：若f:ℝᵈ→ℝ是非负可测函数，且0<m≤M<∞，则不可能在ℝᵈ中几乎所有仿射k-平面P上，f在P上的积分都介于m和M之间。设G=ℤᵏ×{0}ᵈ⁻ᵏ，取m=M=1且f为ℝᵈ中某可测集的指示函数，由此可推出：ℝᵈ中不存在关于群G的可测Steinhaus集，即不存在可测集S⊆ℝᵈ，使得S通过所有T∈O(d)对应的T(G)平铺ℝᵈ。我们还将关于函数线积分大小的不可能性结果推广到平面上带形区域的积分。
英文摘要
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.
