In this paper we show that any convex quasi-homogeneous projective domain whose boundary is everywhere twice differentiable except possibly at a finite number of points is homogeneous.
Kyeonghee Jo. "Homogeneity, quasi-homogeneity and differentiability of domains." Proc. Japan Acad. Ser. A Math. Sci. 79 (9) 150 - 153, Nov. 2003. https://doi.org/10.3792/pjaa.79.150