A simpler definition of a convex function f is f(x) = sup { l(x) | l <= f where l is linear }. If l <= f is linear then E[f(X)] >= E[l(X)] = l(E[X]). Taking th

A simpler definition of a convex function f is f(x) = sup { l(x) | l <= f where l is linear }. If l <= f is linear then E[f(X)] >= E[l(X)] = l(E[X]). Taking the sup shows E[f(X)] >= f(E[X]).