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.

## Relative interior.

roposition

Let C be a nonempty convex set consisting of more then one point.

a. ( Non emptiness of relative interior). The is not empty and . If then there are vectors such that span the subspace parallel to .

b. ( Line segment principle). If and then all point of the line connecting and , except possibly the , belong to .

Let be a vector such that is a subspace. One can choose vectors such that , are linearly independent and the linear span of is . All convex combinations of belong to and also belong to . Hence, the is not empty. We construct as claimed in (a) by taking . Consequently, .

The statement (b) is evident from the picture ( Relative interior ).

Relative interior.

 Notation. Index. Contents.