Conjugate sets

  1. Prove that
  2. Prove that
  3. Prove that if is a ball of radius at some norm with the center in zero, then
  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:
  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 , 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:

  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 as follows:

    , then unit ball with the zero point as the center is the only self conjugate set in .

  20. Find the conjugate set to the ellipsoid:

  21. Let be the subspace of a Euclidian space . Prove that , where - orthogonal complement to .

  22. Let be the set of all dimensional antisymmetric matrices. Show that .