Polynomial hulls and an optimization problem

Abstract

We say that a subset of Cn 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 Bn be the open unit ball in Cn.Suppose K is a C∞ compact manifold in ∂B1 × Cn, n > 1, diffeomorphic to ∂B1 × ∂Bn, each of whose fibers Kz over ∂B1 bounds a strictly hypoconvex connected open set. Let K be the polynomialhull of K. Then we show that K∖K is the union of graphs of analytic vector valued functions on B1. 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 pis a C∞ real-valued function on ∂B1× Cn, we show that the infimum γρ = infƒ∈H ∞ (B1)n ‖ρ(z, ƒ (z))‖∞ is attained by a unique bounded ƒ provided that the set (z, w) ∈ ∂B1 × C n¦ρ(z, w) ≤ γρ has bounded connected strictly hypoconvex fibers over the circle.