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On polyhedra with the local Pompeiu property. (English. Russian original) Zbl 1075.52503
Dokl. Math. 62, No. 1, 69-71 (2000); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 373, No. 4, 448-450 (2000).
From the text: The classical Pompeiu problem about functions with zero integrals over the sets congruent to a given one has been studied by many authors. Of much interest are the local versions of the Pompeiu problem about functions having given integral means and defined on bounded domains.
Let \(R^n\) be the real Euclidean space of dimension \(n\geq 2\) with Euclidean norm \(|\cdot|\), \(\text{ISO} (n)\) be the isometry group of \(R^n\), and \(B_r=\{x\in R^n:|x|<r\}\) be an open ball.
A compact set \(A\subset R^n\) is called a Pompeiu set in the ball \(B_r\) if every locally summable function \(f:B_r\to\mathbb{C}\) for which \[ \int_Af(\lambda x)dx=0\quad\text{at all }\lambda\in\text{ISO}(n): \lambda A\subset B_r \] vanishes almost everywhere. Many compact sets \(A\) are Pompeiu if the size of \(B_r\) is sufficiently large in comparison with the size of \(A\). In earlier papers, the author stated the following problem.
Problem. Given \(A\), find \[ r(A)=\inf \bigl[r>0:A\in{\mathcal P} (B_r)\bigr\}, \] where \({\mathcal P}(B_r)\) is the set of all Pompeiu sets in the ball \(B_r\).
A number of results containing upper estimates for the values of \(r(A)\) have been obtained by Berenstein and Gay. So far, the exact values of \(r(A)\) have only been known in the following cases: (i) for \(A\) being a parallelepiped in \(R^n\), (ii) if \(A=\{x=(x_1,x_2,\dots, x_n) \in R^n:|x|\leq 1\), \(x_n\geq 0\}\) is a half-ball in \(R^n,\) then \(r(A)=\frac {\sqrt 5}{2}\); (iii) if \(n=2\) and \(A\) is the equilateral triangle with side \(a\), then \(r(A)=\frac{a\sqrt 3}{2}\).
In this paper, we evaluate \(r(A)\) for a large class of polyhedra \(A\subset R^n\) (see Theorem 1–3). The technique developed for proving these results allows us to solve one extremal problem related to the Morera theorem and obtain new mean value theorems for various classes of polynomials, which substantially strengthen the results of Flatto.
52A22 Random convex sets and integral geometry (aspects of convex geometry)
28A78 Hausdorff and packing measures
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures