On the compactness and continuity of polytopes

This paper considers the fundamental properties of polytopes in Banach space. A norm is defined for the polytopes and the preliminarily properties are then analyzed. It is proved that the linear summation of two polytopes is still a polytope. For a given compact set, its polytope is compact if and only if the supporting space is with finite dimension. If the dimension of its supporting space is infinite, the polytope is totally bounded. The continuities of the polytopes is analyzed. It is verified it is lower semi-continuous, and the null set of weak inverse is a linear subspace.