Our purpose is to study the
geometry of the unit ball in the space of real measures on the boundary of a finite
Riemann surface that annihilate the analytic functions on that surface with
continuous boundary values. We show that the number of boundary components of
the surface (and hence the topological type) can be determined from the
geometry of this unit ball. More precisely, if the surface has k boundary
components then this unit ball has 2k− 2 “flat faces” of highest possible
dimension. We also get some information on extreme points and conclude
with an example to show that the linear structure of this unit ball depends
not only on the topology of the surface but also on some of its conformal
structure.
