## Seemous 2015 Problem 1

**Problem 1.** Prove that for every the following inequality holds:

## Krein Milman Theorem

Let be a normed vector space and let be a convex set. A point is said to be *extreme point* if

1. Check that is an extreme point if and only if is convex.

2. Let be an extreme point of . Let be elements of and be positive real numbers such that . Prove that if then .

In what follows we assume that is a nonempty compact convex set. A subset is said to be an *extreme set* if is nonempty, closed, and whenever are such that for some then and .

3. Let . Check that is an extreme point if and only if is an extreme set.

The goal of points 4-6 is to prove that every extreme set contains at least one extreme point.

4. Let be an extreme set and let . Set

Prove that is an extreme subset of .

5. Let be an extreme set of . Consider the collection of all extreme sets of contained in , equipped with the following order

Prove that has a maximal element .

6. Prove that is reduced to a single point, and deduce that there exists an extreme point in .

7. Prove that coincides with the closed convex hull of all its extreme points (Krein-Milman Theorem).

*H. Brezis, Functional Analysis, Problem 1*

## Agregation 2001 – Dual of convex sets

Denote a compact convex part of . If is an invertible matrix prove that

where

For an denote .

a) Prove that is a closed interval which is unbounded from above.

b) Denote which is a nonnegative real number. Prove that

Give explicitly and for the following examples:

(i) is the unit disk in ;

(ii) is the square in ;

(iii) is a parallelogram in centered at the origin.

Prove that is convex, compact, contains in its interior and for every we have

Suppose that is symmetric by the origin. Prove that and are norms. What can you say about and ?

Denote the projection on the compact convex set . If and is the hyperplane which passes through and is orthogonal to the line passing through and prove that there is an equation of of the form

for a certain vector such that , and for every point .

Prove that .

## Caratheodory’s Theorem on Convex Hulls

Caratheodory’s theorem states that if a point is in the convex hull of , then there is a subset consisting in at most elements such that lies in the convex hull of .

## Convex functions limit Traian Lalescu 2010

Suppose that the sequence of convex functions converges pointwise to on . Prove that it converges uniformly to on .

*Traian Lalescu student contest 2010, Iasi, Romania*

## Fixed point theorem

Suppose is a compact convex set and is a function satisfying . Prove that has a fixed point ( i.e. with ). Read more…

## Convex function & limit

Suppose is convex and differentiable with . Prove that . Read more…