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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.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

Caratheodory's theorem.

heorem

Let be a non-empty subset of .

1. For every there are linearly independent vectors , such that for some and finite .

2. For every there are vectors , such that for some and the vectors are linearly independent.

Remark

The Caratheodory theorem does not state that might serve as a fixed basis. Indeed, on the picture ( Caratheodory theorem remark ) if the set X is open then for any pair of vectors and from X a point may be found outside of the area span by the positively linear combinations of and .

Caratheodory theorem remark.

Proof

1. The definition of provides that there are some vectors such that If such vectors are linearly dependent then there are numbers We take a linear combination of two equalities and note that for at least one the is positive. Hence, a exists such that all are non negative and for at least one index . Hence, we decreased the number of terms in the sum. We continue this process until are linearly independent.

Proof

2. The definition of provides that there are some vectors such that We consider and restate the above conditions as Therefore, . The first part of the theorem applies and the vectors may be assumed linearly independent. Hence, no all non-zero exist such that Equivalently, We express the from the second equation and substitute it into the first. We obtain the following consequence We conclude that no all non-zero exist such that the above is true. Hence, the are linearly independent.

 Notation. Index. Contents.