Identification of generic polygonal domains by integral-geometric invariants

Abstract

We study a reconstruction problem of planar domains from non-local integral-geometric invariants. We show that a generic polygonal domain, not necessarily convex, is uniquely determined, up to Euclidean isometry, by the interpoint distance distribution (IDD), which, for convex domains, is equivalent to the chord length distribution. Using a boundary representation of the Riesz energy function, we replace the IDD of the domain by an equivalent boundary IDD weighted by the scalar product of the outer unit normals, which we call $ν$-weighted IDD of the boundary. It enables us to reduce the problem to a one-dimensional problem. By analyzing jumps and blow-up terms of the second and third derivatives of the $ν$-weighted IDD of the boundary, we recover the side lengths, their cyclic incidence, and the exterior angles of the polygon. This can be considered as extending Waksman's classical generic reconstruction theorem for convex polygons to non-convex setting, as well as extending generic polygonal reconstruction from directional covariogram data to a one-dimensional invariant in which directional information has been integrated out.