A function is called a convex function if the domain of the function is a convex set and if for all \(x,y \in \boldsymbol{dom} f\) and \(\theta\) with \(0 \leq \theta \leq 1\), we have
\[f(\theta x + (1-\theta) y) \leq \theta f(x) + (1-\theta) f(y)\]Geometrically the above inequality means that the line segment joining the points \((x, f(x))\) and \((y, f(y))\) must lie above the graph of the function \(f\).
The function \(f\) is called strictly convex if there is a strict inequality in the above function whenever \(x \neq y\) and \(0 < \theta < 1\).
The function \(f\) is called concave if \(-f\) is a convex function. Similarly \(f\) is strictly concave if \(-f\) is strictly convex.
In the above equation if there is always a equality, then it is called an [Affine Function] .