A symmetry result for strictly convex domains

TL;DR

This paper proves that a strictly convex domain with smooth boundary in the plane must be a disc if certain integral conditions involving exponential functions and polynomial powers are satisfied for large degrees.

Contribution

It establishes a new symmetry characterization of strictly convex domains based on integral conditions involving exponential and polynomial functions.

Findings

The domain must be a disc if the integral condition holds for all large n.

The result links integral conditions to geometric symmetry.

Provides a new criterion for identifying circular domains.

Abstract

Assume that $D\subset \R^2$ is a strictly convex domain with $C^2-$smooth boundary. {\bf Theorem.} {\em If $\int_De^{ix}y^ndxdy=0$ for all sufficiently large $n$, then $D$ is a disc.}