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.