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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 $A=[-a_1,a_1]\times\cdots\times[-a_n,a_n].$
MSC:
 26B15 Integration of real functions of several variables: length, area, volume 28A99 Classical measure theory 43A99 Abstract harmonic analysis
Keywords:
Pompeiu sets; vanishing integral