Circumscribed sphere of a convex polyhedron

Let K be a convex polyhedron in Rn with non-empty interior, and P1,P2,…,Pm,m≥n+1, are vertices of K. Then K is the union of finite number of n-dimensional simplices {∑i=1n+1tiPji:ti≥0,∑i=1n+1ti=1} for which the convex hull of the (n−1)-dimensional circumscribed sphere of vertices Pj1, Pj2,…,Pjn+1 contains K. The result is applied to solve a linear programming problem concerning the circumscribed sphere of a convex polyhedron.