Lipschitz conditions, strong uniqueness, and almost Chebyshev subspaces of C(X)

AbstractFor a finite dimensional subspace M of C(X), X a compact metric space, it is well known that the (set valued) metric projection PM is (Hausdorff) continuous at any f ϵ C(X) having a unique best approximation from M and is point Lipschitz continuous at any f ϵ C(X) having a strongly unique best approximation from M. The converses of these classical results are studied. It is shown that if f has a unique best approximation and PM is point Lipschitzian at f, then f has a strongly unique best approximation. If M is an almost Chebyshev subspace of C(X), then the converses of both statements above are shown to hold. Using a theorem of Garkavi, the validity of these converses actually characterizes the almost Chebyshev subspaces of C(X)