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}.

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