Convex Radiant Costarshaped Sets and the Least Sublinear Gauge

The paper studies convex radiant sets (i.e. containing the origin) of a linear normed space

and their representation by means of a gauge. By gauge of a convex radiant set

C\subseteq X C ⊆ X

we mean a sublinear function

p:X\to{\bar{\mathbb R}} p : X → R ˉ

such that

C=[p\leq 1] C = [ p ≤ 1 ]

. Besides the most important instance, namely the Minkowski gauge

\mu_C(x)=\inf\{\lambda >0: \,x\in\lambda C\} μ C ( x ) = inf ⁡ { λ > 0 :   x ∈ λ C }

, the set

C C

may have other gauges, which are necessarily lower than

\mu_C μ C

. We characterize the class of convex radiant sets which admit a gauge different from

\mu_C μ 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 C

, that is the kernel (in the sense of Starshaped Analysis) of the complement of

C C

. We also devote some attention to the relation between costarshaped and hyperbolic convex sets.