Extremal problems about the Pompeiu sets with the spherical boundary (Q2880011)

The smallest quantity of the radius of ball in which an arbitrary set to be a Pompeui set, is found. Denote by \(M(n)\) a group of the motions in \({\mathbb R}^n,\) and let \(\text{ Mot\,} (A, B)=\{\lambda\in M(n): \lambda A\subset B\}.\) A compact set \(A\subset {\mathbb R}^n,\) \(n\geq 3,\) is called the Pompeiu set in \(B,\) write \(A\in \text{ Pomp\,}(B),\) iff the condition \(\int\limits_{\lambda A} f(x) dx=0\) for all \(\lambda\in \text{ Mot\,} (A, B)\) implies that \(f=0\) a.e. for every integrable function \(f:B\rightarrow {\mathbb C}.\) Follow, given a set \(A\) we denote \({\mathcal P}(A)=\inf\{R>0: A\in \text{ Pomp\,}{\mathbb B}_R\}\) and \({\mathbb S}_h=\{x\in {\mathbb R}^n: | x| \leq 1, x_n\geq 1-h\}.\) The main result of the paper is the following. Let \(h\in (\sqrt{5}-1, 2),\) then \({\mathcal P}({\mathbb S}_h)=h.\) The result mentioned above has a series of the consequences which are given in the paper, also.