Polynomial hulls and H-infinity control for a hypoconvex constraint

TL;DR

This paper proves two conjectures related to hypoconvex sets in complex analysis, showing that under certain conditions, the polynomial hulls are unions of analytic graphs and establishing uniqueness and smoothness of these graphs.

Contribution

It proves two conjectures of Helton and Marshall, demonstrating the structure and uniqueness of analytic vector-valued functions associated with hypoconvex sets in complex space.

Findings

Polynomial hulls are unions of analytic graphs with boundary in Y.

There is a unique analytic vector-valued function f satisfying p(z,f(z)) ≤ t.

The function f is smooth on T and varies smoothly with parameters.

Abstract

We say that a subset of C^n is hypoconvex if its complement is the union of complex hyperplanes. Let D be the closed unit disk in C, T the unit circle. We prove two conjectures of Helton and Marshall. (See ``Frequency domain design and analytic selections,'' Indiana Univ. Math. J. 39, no. 1 (1990), 157-184.) Let p:T X C^n --> R+ be a smooth function whose sublevel sets have compact hypoconvex fibers over T. Then, with some restrictions on p, if Y is the set where p is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector-valued functions with boundary in Y. Furthermore, let t be the smallest real number such that the set where p is less than or equal to t contains the boundary of the graph of some analytic vector-valued function on the disk. Then there is only one analytic vector-valued function f such that p(z,f(z)) is less than or equal to t for all z in T. We show that f is smooth on T. We also prove that if p varies smoothly with respect to a parameter, so does the unique f just found.