It is well known that the axiom of choice implies the following axiom:
VKM: If E is a locally convex Hausdorff linear topological vector space, every non-empty convex-compact convex subset of E has at least one extreme point.
We show that the axiom VKM implies the following axiom:
WO: Every well-ordered family of non-empty sets has a non-empty product.

This partly solves the following question, raised by Bell and Fremlin (1972), see [B-F72] : Does the axiom VKM imply the axiom of choice?