We introduce the notion of the finite dimensional approximation property (the FDAP) and prove that if a subset X of a linear metric space has the FDAP, then every non-empty convex subset of X is an AR.

As an application we show that every needle point space X contains a dense linear subspace E with the following properties:

(i) E contains a non-empty compact convex set with no extreme points;

(ii) all non-empty convex subsets of E are AR.