Isomorphisms of spaces of continuous affine functions (Q804888)

Let \(K\) and \(S\) be compact convex sets and let \(A(K)\) and \(A(S)\) be the corresponding Banach spaces of continuous affine functions. If the Banach-Mazur distance between \(A(K)\) and \(A(S)\) is less than 2, then under certain geometric conditions, the extreme boundaries of \(K\) and \(S\) are homeomorphic. This extends a result of Amir and Cambern, and has applications to function algebras.