The following statements are consequences of the propositions ( Minimax theorem ) and ( Partial minimization result ).

(Saddle point result 1) Assume that

1. the function is convex and closed,

2. the function is convex and closed,

3. .

Then the minimax equality holds and is nonempty under any of the following conditions.

0. The level sets of the function are compact.

1. The recession cone and the constancy space of the function are equal.

2. The function has the form with being a closed proper convex function and set being given by the linear constraints

and .

3. where are symmetric matrices, is positive semidefinite, is positive definite, where the are positive semidefinite matrixes.

In addition, if (0) holds then is compact.

(Saddle point result 2). Assume that

3. Either or .

Then

(a) If the level sets of functions and are compact then the set of saddle points of is nonempty and compact.

(b) If and then the set of saddle points of is nonempty.

(Saddle point theorem). Assume that

Then the set of saddle points of is nonempty and compact if any of the following conditions are satisfied

1. and are compact.

2. is compact and is nonempty and compact for some and .

3. is compact and is nonempty and compact for some and .

4. and are nonempty and compact for some , and .