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

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

  • If
  • 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 and linear function , is convex