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}.

Read more…

Krein Milman Theorem

October 9, 2012 Leave a comment

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

Read more…

Agregation 2001 – Dual of convex sets

September 7, 2012 Leave a comment

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}.

Read more…

Convex functions limit Traian Lalescu 2010

May 17, 2010 2 comments

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

Read more…

Fixed point theorem

January 12, 2010 Leave a comment

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…

Convex function & limit

January 6, 2010 Leave a comment

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…

Categories: Analysis, Problem Solving Tags: ,
%d bloggers like this: