Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 2 Gauss-Hermite Integration.
 3 Asymptotic expansions.
 4 Monte-Carlo.
 5 Convex Analysis.
 A. Basic concepts of convex analysis.
 B. Caratheodory's theorem.
 C. Relative interior.
 D. Recession cone.
 E. Intersection of nested convex sets.
 F. Preservation of closeness under linear transformation.
 G. Weierstrass Theorem.
 H. Local minima of convex function.
 I. Projection on convex set.
 J. Existence of solution of convex optimization problem.
 K. Partial minimization of convex functions.
 L. Hyperplanes and separation.
 M. Nonvertical separation.
 N. Minimal common and maximal crossing points.
 O. Minimax theory.
 Q. Polar cones.
 R. Polyhedral cones.
 S. Extreme points.
 T. Directional derivative and subdifferential.
 U. Feasible direction cone, tangent cone and normal cone.
 V. Optimality conditions.
 W. Lagrange multipliers for equality constraints.
 X. Fritz John optimality conditions.
 Y. Pseudonormality.
 Z. Lagrangian duality.
 [. Conjugate duality.
 a. Support function.
 b. Infimal convolution.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Infimal convolution.

he following statement is direct consequence of definitions.

Proposition

(Infimal convex function)Let be a convex set. The function is convex.

Definition

(Infimal convolution) The function is called infimal convolution.

Proposition

If are convex functions then is a convex function.

Proof

Apply the result ( Infimal convex function ) to the set .

Proposition

(Duality of infimal convolution and addition). Let be convex functions. Then

Proof

By definition of infimal convolution ( Infimal convolution ) and the duality operation ( Conjugate duality theorem ) we have

Corollary

(Infimal convolution of support functions). Let be non-empty convex sets, . Then

 Notation. Index. Contents.