  Let T be the unit circle in the complex plane C. This paper proves the
existence of analytic structure in a compact subset K of T X C^n, where K has
so-called "lineally convex" or "hypoconvex" fibers over T. It also addresses a
related H-infinity optimization problem. The theorems here remove a number of
unnatural assumptions required in an earlier work by the same author,
"Polynomial hulls and H-infinity control for a hypoconvex constraint." (See
http://www.arxiv.org/abs/math.CV/0001039)
