  Let X be a nonempty convex compact subset of some Haus-dorff locally convex
topological vector space S. The well know Bauer's maximum principle stats that
every convex upper semi-continuous function from X into R attains its maximum
at some extremal point of X. We give some extensions of this result when X is
assumed to be compact metrizable. We prove that the set of all convex upper
semi-continuous functions attaining there maximum at exactly one extremal point
of X is a G $\delta$ dense subset of the space of all convex upper
semi-continuous functions equipped with a metric compatible with the uniform
convergence .
