Convex sets

  1. Show that the set is convex if and only if its intersection with any line is convex.
  2. Show that the convex hull of the set is the intersection of all convex sets containing .
  3. Let is a random variable with a given probability distribution of , where , and . It is said that the probability vector of outcomes of belongs to the probabilistic simplex, i.e. . Determine if the following sets of are convex:
    1. , где означает математическое ожидание заданной функции , т.е.
  4. Prove that if the set is convex, its interior is also convex. Is the opposite true?
  5. Prove that if the set is convex, its closure is also convex. Is the opposite true?
  6. Prove that the set of square symmetric positive definite matrices is convex.
  7. Show that the set of is convex if and only if

  8. Calculate the Minkowski sum of the line segment and the square on the plane, the line segment and the triangle, the line segment and the circle, the line segment and the disk.
  9. Find the minimum value of , at which the set of is convex.
  10. Prove that the set of is convex.
  11. Give an example of two closed convex sets, the sum of which is not closed
  12. Find convex and conical hulls of the following sets: , ,
  13. Show that the set of directions of the strict local descending of the differentiable function in a point is a convex cone.
  14. Prove that :

    is a convex cone.

  15. Find the convex hulls of the following sets:
  16. For an arbitrary set of , let’s say consists of all segments of with the ends of . Is it true that ?
  17. Is the given set a convex polyhedron (could be written in the form of ):
  18. Let is a set of solutions to the quadratic inequality:

    1. Show that if , is convex. Is the opposite true?
    2. Show that the intersection of with the hyperplane defined by the is convex if for some real . Is the opposite true?
  19. Show that the hyperbolic set of is convex. Hint: For it is valid, that with non-negative .
  20. Which of the sets are convex:
    1. Stripe,
    2. Rectangle,
    3. Kleen,
    4. A set of points closer to a given point than a given set that does not contain a point,
    5. A set of points, which are closer to one set than another,
    6. A set of points, , where is convex and is arbitrary.
    7. A set of points whose distance to a given point does not exceed a certain part of the distance to another given point is
  21. Find the conic hull of the set of rank matrix products ?
  22. Let is a cone. Prove that it is convex if and only if a set of is convex.
  23. Let be such that . Is this set convex?
  24. Find the conic hull of the following sets in :
  25. Let is a disk of and is a segment of . How their convex combination with looks like.
  26. Is the next set convex?

  27. Prove that in order for to be a convex cone, it is enough that contains all possible non-negative combinations of its points.
  28. Prove that in order for to be an affine set it is necessary and sufficient that contains all possible affine combinations of its points.
  29. Пусть $S_1, \ldots, S_k$ - произвольные непустые множества в $\mathbb{R}^n$. Докажите, что:
    • $ \mathbf{cone} \left( \bigcup\limits_{i=1}^k S_i\right) = \sum\limits_{i=1}^k \mathbf{cone} \left( S_i\right) $
    • $ \mathbf{conv} \left( \sum\limits_{i=1}^k S_i\right) = \sum\limits_{i=1}^k \mathbf{conv} \left( S_i\right) $
  30. Prove, that the set $S \subseteq \mathbb{R}^n$ is convex if and only if $(\alpha + \beta)S = \alpha S + \beta S$ for all non-negative $\alpha$ and $\beta$
  31. Let is a random variable with a given probability distribution of , where , and . It is said that the probability vector of outcomes of belongs to the probabilistic simplex, i.e. . Determine if the following sets of are convex:

  32. Prove, that ball in (i.e. the following set ) - is convex.
  33. Prove, that if is convex, then . Give an counterexample in case, when - is not convex.