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.

## Nonvertical separation.

iven a space we can separate the last variable and call a hyperplane vertical if its normal vector is of the form . A set for a fixed is called a vertical line.

Proposition

(Nonvertical separation). Let be a nonempty convex subset of that contains no vertical lines. Then:

1. The is contained in a half-space of a nonvertical hyperplane.

2. If then there is a nonvertical hyperplane that separates and .

Proof

1. By contradiction and proposition ( Intersection of half-spaces ), if all half-spaces that surround come from vertical hyperplanes then must have a vertical line.

2. Consider . If is not of the form then we are done. If its is of the form then we use a perturbation on the figure ( Nonvertical separation figure ). First, take any hyperplane from the part (1) of the statement. There are no points of in the part of the space below the broken plane (A,O,D). We perform a slight perturbation the hyperplane (A,B) into that area while maintaining separation from the point .

Nonvertical separation.

 Notation. Index. Contents.