Proximinal subspaces of A(K) of finite codimension

Abstract

We study an analogue of Garkavi′s result on proximinal subspaces of C(X) of finite codimension in the context of the space A(K) of affine continuous functions on a compact convex set K. We give an example to show that a simple‐minded analogue of Garkavi′s result fails for these spaces. When K is a metrizable Choquet simplex, we give a necessary and sufficient condition for a boundary measure to attain its norm on A(K). We also exhibit proximinal subspaces of finite codimension of A(K) when the measures are supported on a compact subset of the extreme boundary.