Circumscribed sphere of a convex polyhedron Original Research Article

Let KK be a convex polyhedron in View the MathML sourceRn with non-empty interior, and P1,P2,…,Pm,m≥n+1P1,P2,…,Pm,m≥n+1, are vertices of KK. Then KK is the union of finite number of nn-dimensional simplices View the MathML source{∑i=1n+1tiPji:ti≥0,∑i=1n+1ti=1} for which the convex hull of the (n−1)(n−1)-dimensional circumscribed sphere of vertices Pj1Pj1, Pj2,…,Pjn+1Pj2,…,Pjn+1 contains KK. The result is applied to solve a linear programming problem concerning the circumscribed sphere of a convex polyhedron. Compact convex sets and convex polyhedra play an important role in many branches. The authors of this paper studied this object to describe the qq-numerical range of a normal matrix [2] and normal operator [1]. Such investigation involved the circumscribed circle of a convex polygon. At first, we have a geometric result for the circumscribed sphere of a convex polyhedron.