Spectral determination of semi-regular polygons

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. The main result of the paper is that, in the class of convex polygons, semi-regular polygons are uniquely determined by just three geometric quantities: the area, the perimeter, and a third quantity depending only on the interior angles of the polygon that appears in the heat trace asymptotics of a polygon. As a consequence of this, we show that semi-regular polygons are spectrally determined, meaning 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 $P$, then $\Omega$ is congruent to $P$.