Numerical range circumscribed by two polygons

We show that, for any 2n+2 distinct points a(1), a'(1), a(2), a'(2),...,a(n+1), a'(n+1) (in this order) on the unit circle, there is an n-by-n matrix A. unique LIP to Unitary equivalence, which has norm one and satisfies the conditions that it has all its eigenvalues in the open unit disc, I-n-A*A has rank one and its numerical range is circumscribed by the two (n+1)-gons a(1)a(2)...a(n+1) and a'(1)a'(2)...a'(n+1). This generalizes the classical result of the existence of a conical Curve Circumscribed by two triangles which are already inscribed on another conical curve. (C) 2004 Elsevier Inc. All rights reserved.