Home > Affine Geometry, Algebra, Analysis > Agregation 2001 – Dual of convex sets

## 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 ${(K^*)^*=K}$.