AnF-space with trivial dual where the Krein-Milman theorem holds

We show that in certain non-locally convex Orlicz function spacesL _ϕ with trivial dual every compact convex set is locally convex and hence the Krein-Milman theorem holds. This complements the example constructed by Roberts of a compact convex set without extreme points inL _p (0<p<1) and answers a question raised by Shapiro.