convex function

Convex functions play an important role in many areas of mathematics.

Let be the objective convex function we want to maximize.

A strictly convex function will have at most one global minimum.

For instance, a (strictly) convex function on an open set has no more than one minimum.

The function has at all points, so f is a convex function.

Any local minimum of a convex function is also a global minimum.

Every norm is a convex function, by the triangle inequality and positive homogeneity.

Every convex function is pseudoconvex, but the converse is not true.

If is a convex function defined on a real interval, then is Schur-convex.

Generalizations to convex functions other than the logarithm is given in Csiszár, 2004.

In mathematics, a concave function is the negative of a convex function.

For each , is a convex function of the tangent space .

The support function is a convex function on .

It follows that h is a convex function.

Let be a convex function with domain .

For every fixed, is a convex function of , while plays the role of constant.

A convex function of a martingale is a submartingale, by Jensen's inequality.

The squared Mahalanobis distance, which is generated by the convex function .

Let be a probability space, X an integrable real-valued random variable and a convex function.

The subderivative and subgradient are generalizations of the derivative to convex functions.

The additive inverse of a convex function is a concave function.

A strongly convex function is also strictly convex, but not vice-versa.

A proper convex function is closed if and only if it is lower semi-continuous.

Invex functions were introduced by Hanson as a generalization of convex functions.

Any convex function on a convex open subset of R is semi-differentiable.