Suppose that to every non-degenerate simplex Δ ⊂ Rn a ‘center’ C(Δ) is assigned so that the following assumptions hold:(i)The map Δ → C(Δ) commutes with similarities and is invariant under the permutations of the vertices of the simplex;(ii)The map Δ → Vol (Δ) C(Δ) is polynomial in the coordinates of the vertices of the simplex. Then C(Δ) is an affine combination of the center of mass CM(Δ) and the circumcenter CC(Δ) of the simplex: (Formula presented.) C(Δ)=tCM(Δ)+(1-t)CC(Δ),where the constant t∈ R depends on the map Δ ↦ C(Δ) (and does not depend on the simplex Δ). 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.
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