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 D be
the closed unit disk in C, T its boundary and let Bn be
the open unit ball in Cn. Suppose E is
a C∞ bounded compact set in T× Cn, n>1,
whose fibers over T are strictly hypoconvex. Let Ê be the
polynomial hull of E. Then we show that Ê \ E is
the union of graphs of analytic vector valued functions on int D.
This result shows that an assumption regarding the deformability of
E in an earlier version of this result (paper 4)
is unnecessary. Next, if
p is a real valued C∞ function on T× Cn, we consider
the minimax I=inff esssupz in T
|p(z,f(z))|,
where f ranges over bounded analytic vector valued functions on D.
We prove that this minimax is attained by a unique bounded f
provided that the set
{(z,w) in T× Cn| p(z,w) < I} possesses the
properties that E does.