ABSTRACT
T h eo rem 1. For every convex polyhedral metric, there exists a unique polyhedron (up to a translation or a translation with a sym m etry) realizing this metric.
Or more precisely: let K be a complex of triangles homeomorphic to a sphere and m its convex metrization. Then there exists a convex polyhe dron p that admits a triangulation forming the complex K and isometric to metrization m, i.e., a triangulation such that the triangles of the metrized complex K and the triangulation of the polyhedron p are isometric. (The triangles on p need not necessarily be planar, but their sides are invariably geodesics on p.)
A convex polygon, which is supposed to be doubly covered and in this sense homeomorphic to a sphere, is also assumed to be a convex polyhedron.
