Remarks on the the circumcenter of mass

TL;DR

This paper characterizes a class of simplex centers that are affine combinations of the centroid and circumcenter, under symmetry and polynomial invariance conditions, extending recent work on the circumcenter of mass.

Contribution

It proves that any such center satisfying the given assumptions must be an affine combination of the centroid and circumcenter, providing a classification result.

Findings

Any center satisfying the assumptions is an affine combination of centroid and circumcenter.

The coefficients of the combination are independent of the specific simplex.

The result generalizes recent studies on the circumcenter of mass of polytopes.

Abstract

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.