A set is convex if line between two points in the set stays in the set.

A function $f$ is convex if all the points above $f$ form a convex set. That is,

- If $f_{′′}(w)≥0,∀w$
- A convex function multiplied by a non-negative constant is convex
- Norms and squared norms are always convex
- The sum of convex functions is convex
- The max of convex functions is a convex function
- The composition of a convex function $g$ and linear function $h$, $g∘h$ is convex