Metric space of unlimited convex sets and unlimited polyhedron

In the paper there is a definition of metric space H(K) the unlimited closed convex subsets of Banach space X, having the same recessive K. There is a Hausdorff metric that is used as a distance. It is established in this paper that the properties of a metric space H(K) are different from the properties of the metric spaceof convex compacts with Hausdorff metric. It is established that the theorem similar to the theorem of approximation of convex compacts polyhedrons is wrong. That is not each element of metric space H(K) can be approximated by the generalized polyhedrons, which are the analogues of the normalpolyhedrons. The paper introduces the concept of a generalized polyhedron in the following way. The set of elements H(0) + K are known as generalized polyhedrons. The criterion of approximation is derived. In order for the element of the space H(K) could be approximated by generalized polyhedrons in the Hausdorff metric it is necessary and sufficient that its basic function was evenly continuous.