zbMATH — the first resource for mathematics

On functions with zero integrals over parallelepipeds. (English. Russian original) Zbl 1064.26500
Dokl. Math. 60, No. 3, 375-376 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 369, No. 4, 444-445 (1999).
A compact set \(A\subset R^n\) is called a Pompeiu set if each locally summable function \(f\:R^n\to C\) for which \(\int_{\lambda A}f(x)dx=0\) for all \(\lambda\in{\text{ISO}}(n)\) vanishes almost everywhere. (\({\text{ISO}}(n)\) is the group of motions of \(R^n\).)
Let \(P(B_r)\) be the class of Pompeiu sets \(A\) for which \(\lambda A\subset B_r\), where \(B_r\) is the sphere of radius \(r\). The problem is to find \(r(A)=\inf\{r>0; A\in P(B_r)\}\). In this paper the author finds the exact value in the case \[ A=[-a_1,a_1]\times\cdots\times[-a_n,a_n]. \]
26B15 Integration of real functions of several variables: length, area, volume
28A99 Classical measure theory
43A99 Abstract harmonic analysis