# Conjugate sets

1. Prove that $S^* = \left(\overline{S}\right)^*$
2. Prove that $\left( \mathbf{conv}(S) \right)^* = S^*$
3. Prove that if $B(0,r)$ is a ball of radius $r$ at some norm with the center in zero, then $\left( B(0,r) \right)^* = B(0,1/r)$
4. Find a dual cone for a monotonous non-negative cone:

5. Find and sketch on the plane a conjugate set to a multi-faceted cone: $S = \mathbf{cone} \{ (-3,1), (2,3), (4,5)\}$
6. Derive the definition of the cone from the definition of the conjugate set.
7. Name any 3 non-trivial facts about conjugate sets.
8. How to write down a set conjugate to the polyhedron?
9. Draw a conjugate set by hand for simple sets. Conjugate to zero, conjugate to the halfline, to two random points, to their convex hull, etc.
10. Give examples of self-conjugate sets.
11. Using a lemma about a cone conjugate, conjugate to the sum of cones and a lemma about a cone, conjugate to the intersection of closed convex cones, prove that cones

are self dual.

12. Find the sets $S^{*}, S^{**}, S^{***}$, if

13. Find conjugate cone for the cone of positive definite (semi-definite) matrices.
14. Find the conjugate cone for the exponential cone:

15. Prove that’s fair for closed convex cones:

16. Find the dual cone for the following cones:

• $K = \{0\}$
• $K = \mathbb{R}^2$
• $K = \{(x_1, x_2) \mid \vert x_1\vert \leq x_2\}$
• $K = \{(x_1, x_2) \mid x_1 + x_2 = 0\}$
17. Find and sketch on the plane a conjugate set to a multifaced cone:

18. Find and sketch on the plane a conjugate set to a polyhedra:

19. Prove that if we define the conjugate set to $S$ as follows:

, then unit ball with the zero point as the center is the only self conjugate set in $\mathbb{R}^n$.

20. Find the conjugate set to the ellipsoid:

21. Let $L$ be the subspace of a Euclidian space $X$. Prove that $L^* = L^\bot$, where $L^\bot$ - orthogonal complement to $L$.

22. Let $\mathbb{A}_n$ be the set of all $n$ dimensional antisymmetric matrices. Show that $\left( \mathbb{A}_n\right)^* = \mathbb{S}_n$.