Conjugate sets
 Prove that
 Prove that
 Prove that if is a ball of radius at some norm with the center in zero, then

Find a dual cone for a monotonous nonnegative cone:
 Find and sketch on the plane a conjugate set to a multifaceted cone:
 Derive the definition of the cone from the definition of the conjugate set.
 Name any 3 nontrivial facts about conjugate sets.
 How to write down a set conjugate to the polyhedron?
 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.
 Give examples of selfconjugate sets.

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.

Find the sets , if
 Find conjugate cone for the cone of positive definite (semidefinite) matrices.

Find the conjugate cone for the exponential cone:

Prove that’s fair for closed convex cones:

Find the dual cone for the following cones:

Find and sketch on the plane a conjugate set to a multifaced cone:

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

Prove that if we define the conjugate set to as follows:
, then unit ball with the zero point as the center is the only self conjugate set in .

Find the conjugate set to the ellipsoid:

Let be the subspace of a Euclidian space . Prove that , where  orthogonal complement to .
 Let be the set of all dimensional antisymmetric matrices. Show that .