Convex function
The function , which is defined on the convex set , is called convex , if:
for any and .
If above inequality holds as strict inequality and , then function is called strictly convex
Examples
 The sum of the largest coordinates
Epigraph
For the function , defined on , the following set:
is called epigraph of the function
Sublevel set
For the function , defined on , the following set:
is called sublevel set or Lebesgue set of the function
Criteria of convexity
First order differential criterion of convexity
The differentiable function defined on the convex set is convex if and only if :
Let , then the criterion will become more tractable:
Second order differential criterion of convexity
Twice differentiable function defined on the convex set is convex if and only if :
In other words, :
Connection with epigraph
The function is convex if and only if its epigraph is convex set.
Connection with sublevel set
If  is a convex function defined on the convex set , then for any sublevel set is convex.
The function defined ob the convex set is closed if and only if for any sublevel set is closed.
Reduction to a line
is convex if and only if is convex set and the function defined on is convex for any , which allows to check convexity of the scalar function in order to establish covexity of the vector function.
Strong convexity
, defined on the convex set , is called strongly convex (strogly convex) on , if:
for any and for some .
Criteria of strong convexity
First order differential criterion of strong convexity
Differentiable defined on the convex set strogly convex if and only if :
Let , then the criterion will become more tractable:
Second order differential criterion of strong convexity
Twice differentiable function defined on the convex set is called strogly convex if and only if :
In other words:
Facts
 is called (strictly) concave, if the function  (strictly) convex.

Jensen’s inequality for the convex functions:
for (probability simplex)
For the infinite dimension case:If the integrals exist and
 If the function and the set are convex, then any local minimum will be the global one. Strong convexity guarantees the uniqueness of the solution.
Operations that preserve convexity

Nonnegative sum of the convex functions:
 Composition with affine function is convex, if is covex
 Pointwise maximum (supremum): If are convex, then is convex
 If is convex on for any : is convex
 If is convex on , then  is convex with
 Let and , where . If and are convex, and is increasing, then is convex on
Other forms of convexity
 Logconvex: is convex; Log convexity implies convexity.
 Logconcavity: concave; not closed under addition!
 Exponentially convex: , for
 Operator convex:
 Quasiconvex:
 Pseudoconvex:
 Discrete convexity: ; “convexity + matroid theory.”