Remarks on the the circumcenter of mass
  Suppose that to every non-degenerate simplex Delta in n-dimensional Euclidean space a `center' C(Delta) is assigned so that the following assumptions hold: (i) The map that assigns C(Delta) to Delta commutes with similarities and is invariant under the permutations of the vertices of the simplex; (ii) The map that assigns Vol(Delta) C(Delta) to Delta is polynomial in the coordinates of the vertices of the simplex. Then C(Delta) is an affine combination of the center of mass and the circumcenter of Delta (with the coefficients independent of 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. 
S. Tabachnikov, E. Tsukerman
math.MG