Spectral determination of semi-regular polygons

TL;DR

This paper proves that semi-regular polygons are uniquely determined by their spectral data among convex piecewise smooth domains, extending spectral geometry results to a broader class of shapes.

Contribution

It establishes that semi-regular polygons are spectrally unique within convex piecewise smooth domains, including those with straight corners.

Findings

Semi-regular polygons are spectrally determined among convex piecewise smooth domains.

Spectral data uniquely identifies semi-regular polygons up to congruence.

The result applies to polygons with possibly one larger angle, generalizing previous spectral uniqueness results.

Abstract

Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible and its angles are all equal but possibly one, which is then larger than the rest. Regular polygons, in particular, are semi-regular. We prove that semi-regular polygons are spectrally determined in the class of convex piecewise smooth domains. Specifically, we show that if $\Omega$ is a convex piecewise smooth planar domain, possibly with straight corners, whose Dirichlet or Neumann spectrum coincides with that of an $n$-sided semi-regular polygon $P_n$, then $\Omega$ is congruent to $P_n$.