Circumscribed sphere of a convex polyhedron

Suppose that K is a convex polyhedron in the n-dimensional Euclidean space R. Suppose that the extreme points of K are P1, P2, . . . , Pm ( m ≥ n + 1). If P is an arbitrary interior point of K, then there exist vertices Pi1 , Pi2 , . . . , Pin+1 of K and a point Q of R n and a positive number r > 0 for which ||Pik − Q|| = r for 1 ≤ k ≤ n + 1 , ||Pj − Q|| ≤ r for 1 ≤ j ≤ m and P is an interior point of the ndimensional simplex with vertices Pi1 , . . . , Pin+1 . This result is related with the q-numerical range of a normal operator. Suppose that T is a bounded linear operator on a complex Hilbert space H and q is a positive number with 0 ≤ q ≤ 1. The q-numerical range Wq(T ) of T is defined as {〈Tx, y〉 : x, y ∈ H, 〈x, x〉 = 〈y, y〉 = 1, 〈x, y〉 = q}.