If $f: R^n\rightarrow R$ is convex and $f(\alpha x)=\alpha f(x), \alpha \geq 0$, show that:
a) $f(x+y)\leq f(x)+f(y)$ for all $x,y\in R^n$
b) $f(0)\geq 0$
c) $f(-x)\geq -f(x)$ for all $x\in R^n$
d) $f(\alpha_1 x_1 +...+\alpha_m x_m)\leq \alpha_1f(x_1)+...+\alpha_mf(x_m)$ for all $\alpha_k>0, x_k\in R^n, k=1, .., m$
Can you help me to prove these inequalities? I apologize for writing them all in one question, but I can't write more questions here today and I need solutions for tomorrow morning.
If your function is continuous. We have for $t \in (0,1)$ \begin{equation} f(tx+(1-t)x) \le tf(x) + (1-t)f(y) \le tf(x) +f(y) \end{equation} Doing $t $ tends to one we obtain a).
To b) $f(0) = f(0x) =0f(x) =0$.
to c) $0 = f(x-x) \le f(x) +f(-x)$
To d) a do simple indction of a)