### Archive

Posts Tagged ‘convex’

## Seemous 2015 Problem 1

March 11, 2015 1 comment

Problem 1. Prove that for every ${x \in (0,1)}$ the following inequality holds:

$\displaystyle \int_0^1 \sqrt{1+\cos^2 y}dy > \sqrt{x^2+\sin^2 x}.$

## Krein Milman Theorem

Let ${E}$ be a normed vector space and let ${K \subset E}$ be a convex set. A point ${a \in K}$ is said to be extreme point if

$\displaystyle tx+(1-t)y\neq a \ \ \forall t\in (0,1),\ \forall x,y \in K,\ x \neq y.$

1. Check that ${a \in K}$ is an extreme point if and only if ${K \setminus \{a\}}$ is convex.

2. Let ${a}$ be an extreme point of ${K}$. Let ${x_1,..,x_n}$ be elements of ${K}$ and ${\alpha_i,\ i=1,n}$ be positive real numbers such that ${\alpha_1+..+\alpha_n=1}$. Prove that if ${\alpha_1x_1+...+\alpha_nx_n=a}$ then ${x_i=a, \ i=1..n}$.

In what follows we assume that ${K \subset E}$ is a nonempty compact convex set. A subset ${M \subset K}$ is said to be an extreme set if ${M}$ is nonempty, closed, and whenever ${x,y \in K}$ are such that ${tx+(1-t)y \in M}$ for some ${t \in (0,1)}$ then ${x \in M}$ and ${y \in M}$.

3. Let ${a \in K}$. Check that ${a}$ is an extreme point if and only if ${\{a\}}$ 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 ${A \subset K}$ be an extreme set and let ${f \in E^*}$. Set

$\displaystyle B=\left\{ x \in A : \langle f,x \rangle =\max_{y \in A} \langle f,y \rangle \right\}.$

Prove that ${B}$ is an extreme subset of ${K}$.

5. Let ${M}$ be an extreme set of ${K}$. Consider the collection ${\mathcal{F}}$ of all extreme sets of ${K}$ contained in ${M}$, equipped with the following order

$\displaystyle A\leq B \text{ if } B \subset A.$

Prove that ${\mathcal{F}}$ has a maximal element ${M_0}$.

6. Prove that ${M_0}$ is reduced to a single point, and deduce that there exists an extreme point in ${M}$.

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

H. Brezis, Functional Analysis, Problem 1

Categories: Functional Analysis

## Agregation 2001 – Dual of convex sets

Denote ${K}$ a compact convex part of ${\Bbb{R}^n}$. If ${A}$ is an invertible ${n \times n}$ matrix prove that

$\displaystyle (A(K))^* =(A^t)^{-1}(K),$

where

$\displaystyle K^*=\{ y \in \Bbb{R}^n : \forall x \in K, \langle x,y \rangle \leq 1\}$

For an ${x \in \Bbb{R}^n}$ denote ${I_x=\{ \lambda \in \Bbb{R}_+ : x \in \lambda K \}}$.

a) Prove that ${I_x}$ is a closed interval which is unbounded from above.

b) Denote ${j_K(x)=\inf I_x}$ which is a nonnegative real number. Prove that

$\displaystyle x \in K \Leftrightarrow j_K(x)\leq 1 \text{ and } x \in \partial K \Leftrightarrow j_K(x)=1.$

Give explicitly ${K^*, j_K}$ and ${j_{K^*}}$ for the following examples:

(i) ${K}$ is the unit disk in ${\Bbb{R}^2}$;

(ii) ${K}$ is the square ${-1 \leq x_1,x_2 \leq 1}$ in ${\Bbb{R}^2}$;

(iii) ${K}$ is a parallelogram in ${\Bbb{R}^2}$ centered at the origin.

Prove that ${K^*}$ is convex, compact, contains ${O}$ in its interior and for every ${y \in \Bbb{R}^n}$ we have

$\displaystyle j_{K^*}(y)=\max \{ \langle x,y \rangle |x \in K\}.$

Suppose that ${K}$ is symmetric by the origin. Prove that ${j_K}$ and ${j_{K^*}}$ are norms. What can you say about ${(R^n,j_K)}$ and ${(R^n,j_{K^*})}$?

Denote ${p_K}$ the projection on the compact convex set ${K}$. If ${a \notin K}$ and ${H}$ is the hyperplane which passes through ${p_k(a)}$ and is orthogonal to the line passing through ${a}$ and ${p_k(a)}$ prove that there is an equation of ${H}$ of the form

$\displaystyle H=\{x \in \Bbb{R}^n |\langle x,u \rangle =1\}$

for a certain vector ${u \in \Bbb{R}^n}$ such that ${\langle a,u \rangle >1}$, and for every point ${x \in K}$ ${\langle x,u\rangle \leq 1}$.

Prove that ${(K^*)^*=K}$.

## Caratheodory’s Theorem on Convex Hulls

Caratheodory’s theorem states that if a point ${x \in \Bbb{R}^d}$ is in the convex hull of ${\Omega \subset \Bbb{R}^d}$, then there is a subset ${P \subset \Omega}$ consisting in at most ${d+1}$ elements such that ${x}$ lies in the convex hull of ${P}$.

## Convex functions limit Traian Lalescu 2010

Suppose that the sequence of convex functions $(f_n)$ converges pointwise to $0$ on $[0,1]$. Prove that it converges uniformly to $0$ on $[0,1]$.
Traian Lalescu student contest 2010, Iasi, Romania

## Fixed point theorem

Suppose $K \subset \mathbb{R}^n$ is a compact convex set and $f: K \to K$ is a function satisfying $\| f(x)-f(y) \| \leq \|x -y\|,\ \forall x,y \in K$. Prove that $f$ has a fixed point ( i.e. $\exists x \in K$ with $f(x)=x$). Read more…
Suppose $f: [0,\infty) \to \mathbb{R}$ is convex and differentiable with $\lim\limits_{x\to \infty} \frac{f(x)}{x}=\ell \in \mathbb{R}$. Prove that $\lim\limits_{x \to \infty} f^\prime (x)=\ell$. Read more…