The Geometry of Representing Measures and their Critical Values

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 2g 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.