## 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 .