Convex functions

First of all let us define what is called convexity. A region $R$ is called convex if given any two points $x,y\in R$ we have

$\alpha x+(1-\alpha )y\in R$, where $\alpha \in [0,1]$.

A function is called convex if it satisfies following property for all $x$ and $y$ in the domain.

$f(\alpha x+(1-\alpha )y)$ ≤ $\alpha f(x)+(1-\alpha )f(y)$, where again $\alpha \in [0,1]$.

--202.141.40.24 17:18, 16 Aug 2004 (UTC)