Proof Mining - Mathematics, Algorithms and Proofs

Proof Mining
The nonseparable/noncompact case
Proposition 5 Let (X, · ) be a strictly convex normed space
and C ⊆ X a convex subset. Then any point x ∈ X has at most
one point c ∈ C of minimal distance, i.e. x − c =dist(x, C).
Hence: if X is separable and complete and provably strictly convex
and C compact, then one can extract a modulus of uniqueness.
Observation: compactness only used to exract uniform bound
on strict convexity (= modulus of uniform convexity) from
proof of strict convexity.
Ulrich Kohlenbach 4