Infinity cones and functions in real vector spaces: An existence result

In the real vector space setting, we prove that any apex-containing cone whose apex is the null vector is the infinity cone of some set. Moreover, we introduce an infinity mapping notion such that any 1-homogeneous extended-real-valued function not identically equal to +infinity coincides with the infinity mapping to some extended-real-valued function. These results heavily rely on the properties of two newly introduced objects, the radially anticonvex sets and the radially anticonvex functions.