A local version of the Pompeiu problem for regular simplex.

Иванисенко Н. С. A local version of the Pompeiu problem for regular simplex. // Taurida Journal of Computer Science Theory and Mathematics, – 2016. – T.15. – №2. – P. 56-
logo DOI https://doi.org/10.37279/1729-3901-2016-15-2-56-67

Let $V$ be a domain in $R^{n}$, $n>2$. A set A is a regular simplex, whose edge is $\sqrt{2}$, in four-dimensional space. Some problems about functions is locally integrable on a set $V$ with vanishing integrals over all images $\lambda A\subset V$ , $\lambda \in M(n)$ of a fixed compact set $A\subset R^{n}$ are studied in the present paper. If the only function is locally integrable on a set $V$ and satisfying this condition is $f=0$ then the set $A$ is called a Pompeiu set in $V$.
Extremely interesting are local versions of the Pompeiu problem, when a function f is defined on a bounded domain $V\subset R^{n}$ and $\int_{\lambda A}f(x)=0$ is required to hold only when $\lambda A\subset V$. In this case the object is to determine conditions on the set $A$ under which have the equality $\int_{\lambda A}f(x)=0$ implies that $f=0$ on $V$.
We will say that a compact set $A\subset R^{n}$ has the local Pompeiu property with respect to the domain $V$ if every function f is locally integrable on a set $V$ have the equality $\int_{\lambda A}f(x)=0$, for all $\lambda A\subset V$, $\lambda \in M(n)$ vanishes almost everywhere in $V$. Such set $A$ is also called a Pompeiu set in $V$. We will denote by $Pomp(V )$ the collection of all Pompeiu sets in the domain $V$.
Of considerable interest is the case when $V$ is the ball $B_{r}\subset R^{n}$, $r>r^{∗}(A)$ (where $r^{∗}(A)$ is the radius of the smallest closed ball containing the set $A$). One can in this case show that for a broad class of sets $A$ the condition $A\in Pomp(B_{r})$ occurs when the size of $B_{r}$ is sufficiently large compared with $A$. The following problem arises in this connection. The following problem arises in this connection.
Problem. Let $A\subset R^{n}$ be a compact set such that $A\in Pomp(B_{r})$ for some $r>r^{∗}(A)$. Find $R(A)=infr>r^{∗}(A):A\subset Pomp(B_{r})$ and investigate when the value $R(A)$ is attainable, that is, $A\subset Pomp(B_{r})$ for $r=R(A)$.
The questions concerning the local version Pompeiu’s problem are investigated in this paper. The case, under considerations is investigation of a regular simplex in the fourth dimension space. A number of results similar to Stokes’s formula are obtained, which allow to calculate integral from some differential operator, which working on set functions though values, similar to integral to a subset or border of a simplex of smaller dimension. In particular, the case when these subset are faces and volume figures of the simplex is considered. Some estimates Pompeiu’s radius were obtained earlier. In this paper estimates are considerably refined. These formulas help to improve the existing evaluation Pompeiu’s radius.
Also we consider the problem about minimal radius of a ball on which $A$ is a Pompeiu’s set. A assessment Pompeiu’s radius, for this regular simplex, have received.

Keywords: a local version Pompeiu’s problem, Pompeiu’s radius, regular simplex, locally integrable function, four-dimensional space.