Simplicial vertex-normal duality with applications to well-centered simplices

We study the relation between the set of n + 1 vertices of an n-simplex S having S ⁿ⁻¹ as circumsphere, and the set of n + 1 unit outward normals to the facets of S. These normals can in turn be interpreted as the vertices of another simplex Ŝ that has S ⁿ⁻¹ as circumsphere. We consider the iterative application of the map that takes the simplex S to Ŝ, study its convergence properties, and in particular investigate its fixed points. We will also prove some statements about well-centered simplices in the above context.