Convexity

Let . Then is convex if the all line segments between any two points in are contained in , that is, if for every and we have

Let be convex and . Then is convex if for every and we have Furthermore, is strictly convex if it also holds that whenever and , the inequality above is strict.

Let be convex and let be continuously differentiable. Then is convex if and only if for all .

Let be convex with a nonempty interior and let be twice continuously differentiable.

1. is convex if and only if is positive semidefinite for all , where is the HessianHessian of at .
2. If is positive definite for all , then is strictly convex.