Boundary-Rigidity of Projective Metrics and the Geodesic X-Ray Transform

Abstract

We prove that given a compact convex non-empty domain {mathcal {M}} in the plane, a function delta :partial mathcal Mtimes partial {mathcal {M}}rightarrow {mathbb {R}}_+ can be extended to a projective metric d on {mathcal {M}} if and only if delta (P,R)+delta (Q,S)-delta (P,S)-delta (Q,R)>0 for any convex quadrangle Box (PQRS) inscribed in partial {mathcal {M}}. Moreover, this extension is unique.