Convex Radiant Costarshaped Sets and the Least Sublinear Gauge

\def\R{\bar{\mathbb R}} The paper studies convex radiant sets (i.e. containing the origin) of a linear normed space $X$ and their representation by means of a gauge. By gauge of a convex radiant set $C\subseteq X$ we mean a sublinear function $p:X\to\R$ such that $C=[p\leq 1]$. Besides the most important instance, namely the Minkowski gauge $\mu_C(x)=\inf\{\lambda >0: \,x\in\lambda C\}$, the set $C$ may have other gauges, which are necessarily lower than $\mu_C$. We characterize the class of convex radiant sets which admit a gauge different from $\mu_C$ in two different way: they are contained in a translate of their recession cone or, equivalently, they are costarshaped, that is complement of a starshaped set. We prove that the family of all sublinear gauges of a convex radiant set admits a least element and characterize its support set in terms of polar sets. The key concept for this study is the outer kernel of $C$, that is the kernel (in the sense of Starshaped Analysis) of the complement of $C$. We also devote some attention to the relation between costarshaped and hyperbolic convex sets.