1. Let us have two different points . Prove that the set of points which in the Euclidean norm are closer to the point than to make up a half-space. Is this true for another norm?
  2. Find if ,
  3. Find if ,
  4. Find if ,
  5. Illustrate the geometric inequality that connects , from which it follows that is a projection of the point onto a convex set of .
  6. For which sets does the projection of the point outside this set exist? Uniquie?
  7. Find if
  8. Find if ,
  9. Let be a closed set, and be a point not lying in it. Show that the projection in norm will be unique, while in norm this statement is not valid.
  10. Find the projection of the matrix on a set of matrices of rank . In Frobenius norm and spectral norm.
  11. Find a projection of the matrix on a set of symmetrical positive semi-definite matrices of . In Frobenius norm and the scalar product associated with it.
  12. Find the projection of point onto the set in norm. Consider the different positions of .
  13. Find , if .
  14. Prove that projection is a nonexpansive operator, i.e. prove, that if is nonempty, closed and convex set, then for any