Convex function: Let X \subset \mathbb{R}^n be a convex set. Assume that the function f: X \rightarrow \mathbb{R} is continuous, then f is said to be convex if f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) for all x_1, x_2 \in X and for all t \in [0,1] . If \leq is replaced with a strict inequaltiy in the definition, we demand x_1 \neq x_2 and t\in(0,1) then f is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting f(x_1) and f(x_2) , the value of the function on the interval [x_1,x_2] is always below the line as illustrated below.