Exercise 28, § 2

Let $f$ be a convex function defined over the convex subset $A$ of a vector space $E$. a) Show that if $A$ is absorbent and $f$ is non-constant then $f$ cannot attain its upper bound in $A$ at the point 0. b) Show that the subset of points of $A$, at which $f$ attains its lower bound in $A$, is convex.