Geometrical and analytical characterizations of positively homogeneous functions

The article provides an overview of results related to the positively homogeneous functions and the spaces formed by various classes of these functions. In the paper we review characteristic properties of positively homogeneous functions of a number of mostly used subspaces of the space of such functions. As such subspaces we consider the subspace of continuous functions, the subspace of Lipschitz functions, the subspace of difference sublinear functions and the subspace of piecewise functions. We give a description of algebraic structures with which endowed these subspaces and establish relationships between them. Along with known results we present a number of new ones with their proofs.