I’m reading Emil Artin’s introduction on the Gamma function, he proved in Theorem 1.5 that “a function is convex, if and only if, it is continuous and weakly convex”.
The definition Artin used is:
The difference quotient: $\varphi(x_1,x_2)=\frac{f(x_1)-f(x_2)}{x_1-x_2}=\varphi(x_2,x_1)$.
$f(x)$ is callled convex (on the interval $(a,b)$) if, for every number $x_3$ of our interval, $\varphi(x_1,x_3)$ is a monotonically increasing function of $x_1$.
We shall call a function defined on an interval weakly convex if it satisfies the inequality $f(\frac{x_1+x_2}2)\leqslant \frac12(f(x_1)+f(x_2))$ for all $x_1, x_2$ of the interval.
I wonder, if there is a weakly convex function that is not convex?
Best Answer
There exist additive discontinuous maps on $\mathbb R$ If f is such a map then $f(\frac {x+y} 2) =\frac {f(x)+f(y)} 2$ and f is not convex because convex functions are continuous.