We know that in a normed linear space the gauge of the unit closed ball is the corresponding norm. Is it possible to have a nonconvex absorbibg set in a vector space whose gauge is a norm.
We know that gauge of a convex set is sublinear. I am trying to find a nonconvex set in a vector space whose gauge is sublinear. Any help is appreciated.
Yes, there are such sets. Take, for example, the subset of the real line $C = [-1, 1] \setminus [0.1, 0.2]$. Then $\gamma_C$ is the standard Euclidean norm on $\Bbb{R}$.