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Proximinal subspaces of \(A(K)\) of finite codimension. (English) Zbl 1036.41014

Summary: 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.

MSC:

41A50 Best approximation, Chebyshev systems
46B20 Geometry and structure of normed linear spaces
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