On unique determination of polyhedral sets

TL;DR

This paper develops geometric methods to establish the uniqueness of polyhedral sets, such as scatterers, using minimal measurements, relying on unique continuation and reflection principles without extra PDE assumptions.

Contribution

It introduces a simplified geometric framework for proving uniqueness of polyhedral sets, applicable to various PDEs and boundary conditions, unifying and extending existing results.

Findings

Unified geometric approach for uniqueness proofs

Minimal measurement requirements established

Applicable to multiple PDE and boundary condition scenarios

Abstract

In this paper we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique continuation and a suitable reflection principle are enough to proceed with the constructions, without any other assumption on the underlying partial differential equation or the boundary condition. We also aim to keep the geometric constructions and their proofs as simple as possible. To illustrate the applicability of this theory, we show how several uniqueness results present in the literature immediately follow from our arguments. Indeed we believe that this theory may serve as a roadmap for establishing similar uniqueness results for other partial differential equations or boundary conditions.