The minimum of the upper convex density of the product of the Cantor set with itself

Let C×C be the product of the Middle Third Cantor set with itself. We prove that the minimum of the upper convex density of C×C is achieved at the vertex, that is, Dcs¯(C×C,V)≤Dcs¯(C×C,x),∀x∈C×C, where VV is a vertex of C×C.