Polynomial hulls and an optimization problem

Abstract

Abstract. We say that a subset of C n is hypoconvex if its complement is the union of complex hyperplanes. We say it is strictly hypoconvex if it is smoothly bounded hypoconvex and at every point of the boundary the real Hessian of its defining function is positive definite on the complex tangent space at that point. Let B n be the open unit ball in C n . Suppose K is a C ∞ compact manifold in ∂B 1 × C n , n > 1, diffeomorphic to ∂B 1 × ∂B n , each of whose fibers K z over ∂B 1 bounds a strictly hypoconvex connected open set. Let K be the polynomial hull of K. Then we show that K \ K is the union of graphs of analytic vector valued functions on B 1 . This result shows that an unnatural assumption regarding the deformability of K in an earlier version of this result is unnecessary. Next, we study an H ∞ optimization problem. If ρ is a C ∞ real-valued function on ∂B 1 × C n , we show that the infimum γ ρ = inf f ∈H ∞ (B 1 ) n ρ(z, f (z)) ∞ is attained by a unique bounded f provided that the set {(z, w) ∈ ∂B 1 × C n |ρ(z, w) ≤ γ ρ } has bounded connected strictly hypoconvex fibers over the circle. §1 Introduction and results. The purpose of this work is to strengthen the results of [Wh] and [V] regarding the existence of analytic structure in a class of polynomial hulls and the solution of an H ∞ optimization problem.For w ∈ C n let w j denote the j th coordinate of w and let |w| denote the standard Euclidean norm. If Y is a compact set in C n , then the polynomial (convex) hull Y of Y is given by Y = {z ∈ C n |P (z)| ≤ sup w∈Y |P (w)| for all polynomials P on C n }.