Convex radiant costarshaped sets and the least sublinear gauge (Q2841068)

The aim of this paper is to study how a closed, convex, radiant subset \(C\) of a real normed linear space \(X\) can be described as a sublevel set \(\{x:p(x)\leq 1\}\) of a sublinear function \(p:X\rightarrow \mathbb {\overline R}\), which is called the gauge of \(C\).NEWLINENEWLINEA subset \(A\) of a real normed space \(X\) is called radiant if \(x\in A,t\in [0,1]\) imply that \(tx\in A\). A subset \(B\) of \(X\) is called coradiant if its complement is radiant.NEWLINENEWLINEThe author first studies sublinear gauges and characterizes convex cones as those convex radiant sets which admit improper gauges. Then he concentrates on the conditions under which the Minkowski gauge is not minimal and proves that \(C\) admits a gauge lower than its Minkowski functional \(\mu_C\) if and only if its outer kernel is nonempty. Next, existence of the least sublinear gauge for all convex radiant sets is proved. Finally, the relations between hyperbolic and costarshaped convex radiant sets is studied.