2026/08/03 by Jun O'Hara
Mathematics · #math.MG #msc:53C65 #msc:52A22
25 pages, 15 figures
arxiv created 2026/08/03 · arxiv updated 2026/08/04
We study the reconstruction of planar polygonal domains from non-local integral-geometric invariants, namely the interpoint distance distribution and the Riesz energy function (Brylinski beta function), together with the chord length distribution in the convex case. We show that a generic polygonal domain is uniquely determined, up to Euclidean isometry, by its Riesz energy function or, equivalently, by its interpoint distance distribution. For convex domains this also yields reconstruction from the chord length distribution, extending Waksman's classical generic reconstruction theorem beyond the convex setting. The proof uses a boundary interpoint distance distribution weighted by the scalar product of the outer unit normals. This weighted distribution is equivalent to the Riesz energy function. By analyzing jumps and blow-up terms of its second and third derivatives at critical lengths, we recover the side lengths, their cyclic incidence, and the exterior angles of the polygon. In an appendix we obtain complementary identification results for regular polygonal domains when the competing domain is convex or has the same number of sides.