Remarks on the circumcenter of mass

Suppose that to every non-degenerate simplex ∆ ⊂ Rn a ‘center’ C(∆) is assigned so that the following assumptions hold: (i) The map ∆ → C(∆) commutes with similarities and is invariant under the permutations of the vertices of the simplex; (ii) The map ∆ → Vol(∆)C(∆) is polynomial in the coordinates of the vertices of the simplex. Then C(∆) is an affine combination of the center of mass CM(∆) and the circumcenter CC(∆) of the simplex: C(∆) = tCM(∆) + (1− t)CC(∆), where the constant t ∈ R depends on the map ∆ 7→ C(∆) (and does not depend on the simplex ∆). The motivation for this theorem comes from the recent study of the circumcenter of mass of simplicial polytopes by the authors and by A. Akopyan.