Any convex cone has an accumulation point in the base by the action of its automorphism group. In this paper, we prove the converse of this statement, more precisely, a convex domain $\Omega$ with a face $F$ of codimension 1 is a cone over $F$ if there is an Aut($\Omega$)-orbit accumulating at a point of $F$.
Kyeonghee Jo. "A characterization of convex cones." Proc. Japan Acad. Ser. A Math. Sci. 84 (10) 175 - 178, December 2008. https://doi.org/10.3792/pjaa.84.175