New results on the Pompeiu problem

Abstract

Let p N ( w ) = ∑ k = 0 N a k w k {p_N}(w) = \sum \nolimits _{k = 0}^N {{a_k}{w^k}} , w ∈ C w \in \mathbb {C} , N ∈ N N \in \mathbb {N} , be a polynomial with complex coefficients. In this paper we prove that if D ⊂ R 2 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 i s ) x(s) = {x_1}(s) + i{x_2}(s) = {p_N}({e^{is}}) , s ∈ [ − π , π ] s \in [ - \pi ,\pi ] , then D D has the Pompeiu property unless N = 1 N = 1 and p 1 ( w ) = a 1 w + a 2 {p_1}(w) = {a_1}w + {a_2} in which case D 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.