We investigate the geometric structure of the unit ball of the Marcinkiewicz sequence space mΨ, giving characterisations of its real and complex extreme points and of the exposed points in terms of the symbol Ψ. Using our knowledge of the geometry of m0Ψ we then give necessary and sufficient conditions for a subset of m0Ψ to be a boundary for Au(Bm0Ψ), the algebra of functions which are uniformly continuous on m0Ψ and holomorphic on the interior of m0Ψ . We show that it is possible for the set of peak points of Au(Bm0Ψ) to be a boundary for Au(Bm0Ψ) yet for Au(Bm0Ψ) not to have a Silov boundary in the sense of Globevnik.