The Geometry of Representing Measures and their Critical Values

Abstract

The author [1] has recently described a smooth parametrization of the convex set of representing measures M₀ for evaluation of analytic functions at a point q₀ in a g holed planar domain with analytic boundary. Here this parametrization is used to provide answers to three questions from the literature concerning the geometry of M₀ and the nature of the critical values of elements in M₀. First, in answer to a question of Nash [5] it is shown that for domains with g > 2 holes the set M₀ is not strictly convex. Further the details of a class of symmetric examples are completed. In these examples it is shown that M₀ is the span of 2^g isolated extreme points answering a question of Nash [5]. Finally, in answer to a question of Sarason [6] it is shown that elements in M₀ can have “double” critical values on the boundary of the domain.