On the One-Dimentional Pompeiu Problem

TL;DR

This paper studies the Pompeiu property for finite unions of intervals on the real line, providing conditions for when such sets have the property under isometric and translation images, revealing the problem's complexity.

Contribution

It offers a necessary and sufficient condition for two-interval unions to have the Pompeiu property under isometries, and explores the complexity for three-interval unions.

Findings

No set of the studied kind has the Pompeiu property under translation images.

A clear criterion is established for two-interval unions to have the property.

Examples illustrate the complexity of the problem for three-interval unions.

Abstract

We investigate the Pompeiu property for subsets of the real line, under no assumption of connectedness. In particular we focus our study on finite unions of bounded (disjoint) intervals, and we emphasize the different results corresponding to the cases where the function in question is supposed to have constant integral on all isometric images, or just on all the translation-images of the domain. While no set of the previous kind enjoys the Pompeiu property in the latter sense, we provide a necessary and sufficient condition in order a union of two intervals to have the Pompeiu property in the former sense, and we produce some examples to give an insight of the complexity of the problem for three-interval sets.