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 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.

## Support function. efinition

The indicator function of a convex set is a function of the form The support function is the conjugate of the indicator function.

According to the definition (Support function)

Note that is a positively homogenous function of . Suppose is some proper convex positively homogenous function. Consider the conjugate By the positive homogeneity . Consequently, for any , Hence, is either 0 or . Introduce the set Such set is a . Indeed, if then and 0 is reached by scaling with . Then . On the other hand if then and is reached by scaling.

Summary

(Convex homogenous function property). If is a proper convex positively homogenous function then and The last part of the summary follows from the result ( Conjugate duality theorem ). We established one-to-one correspondence between convex sets and proper convex positively homogenous functions.

 Notation. Index. Contents.