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Extremal problems on Pompeiu sets. (English. Russian original) Zbl 0957.53042
Sb. Math. 189, No. 7, 955-976 (1997); translation from Mat. Sb. 189, No. 7, 3-22 (1998).
A compact subset \(K\) of \(\mathbb{R}^n\), \(n\geq 2\), is said to have the Pompeiu property if, whenever a function \(f\in C(\mathbb{R}^n)\) averages to zero on every set \(\sigma(k)\), \(\sigma\in M(n)\), the group of rigid motions of \(\mathbb{R}^n\), then \(f\equiv 0\). Since no ball has the Pompeiu property, one considers the same question for pairs \(B_1,B_2\) of closed balls of different radii \(r_1, r_2\), respectively. It is known that there is a countable exceptional set \(P_n\) such that the pair \(B_1,B_2\) will have the Pompeiu property if and only if \(r_1/r_2 \notin P_n\). We refer to [L. Zalcman, Am. Math. Monthly 87, 161-175 (1980; Zbl 0433.53048)] for an introduction to this kind of problem and to the bibliography compiled by L. Zalcman in [NATO ASI Ser., Ser. C 365 185-194 (1992; Zbl 0830.26005), updates available from the author]. Typically this type of problem can be reduced to a problem of harmonic analysis of ideals in the space of distributions of compact support in \(\mathbb{R}^n\). This depends in an essential way on the fact that one is posing the problem in the whole space. In [J. Anal. Math. 52, 133-166 (1989; Zbl 0668.30037)], R. Gay and this reviewer showed that one could prove local versions of this theorem, that is, \(\mathbb{R}^n\) replaced by a ball \(B\). Clearly, there is an obvious minimum size condition on \(B\), namely that the radius of \(r\) of \(B\) is sufficiently large so that \(K\) or a translate of \(K\), fits into \(B\). It turns out for the case of two balls \(B_1,B_2\) of respective radii \(r_1,r_2\), one needs that \(r_1+r_2<r\). There are similar theorems for other \(K\). In the present paper the author gives sharp bounds for this type of size conditions.

53C65 Integral geometry
43A85 Harmonic analysis on homogeneous spaces
26B15 Integration of real functions of several variables: length, area, volume
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