Are the stars of vertices in a barycentric subdivision of a (standard) n-simplex convex?
I assume that it must be valid, but concretely to prove it I come again and again to obstacles. Approaches so far were to work with the cone. For example, the star of the "inner" vertex is always the simplex itself. If one considers (n-1)-simplexes of the boundary, one can take the inner vertices here. Then the cone of the star restricted to the boundary of this inner corner results in the star. Since cones of convex sets are convex, the assumption after induction is valid here.
But for the other vertices, as the dimension increases, it becomes more and more difficult to move forward. Maybe someone has a good idea.
