New results on the Pompeiu problem

Abstract:

Let ${p_N}(w) = \sum \nolimits _{k = 0}^N {{a_k}{w^k}}$, $w \in \mathbb {C}$, $N \in \mathbb {N}$, be a polynomial with complex coefficients. In this paper we prove that if $D \subset {\mathbb {R}^2}$ is a simply-connected bounded open set whose boundary is a closed, simple curve parametrized by $x(s) = {x_1}(s) + i{x_2}(s) = {p_N}({e^{is}})$, $s \in [ - \pi ,\pi ]$, then $D$ has the Pompeiu property unless $N = 1$ and ${p_1}(w) = {a_1}w + {a_2}$ in which case $D$ is a disk. This result supports the conjecture that modulo sets of zero two-dimensional Lebesgue measure, the disk is the only simply-connected, bounded open set which fails to have the Pompeiu property.