$k-$smoothness on polyhedral Banach spaces

Abstract

We characterize $k-$smoothness of an element on the unit sphere of a finite-dimensional polyhedral Banach space. Then we study $k-$smoothness of an operator $T\in \mathbb{L}({\ell }_{\infty }^n,\mathbb{Y}),$ where $\mathbb{Y}$ is a two-dimensional Banach space with the additional condition that $T$ attains norm at each extreme point of $B_{{\ell }_{\infty }^n}.$ We also characterize $k-$smoothness of an operator defined between ${\ell }_{\infty }^3$ and ${\ell }_1^3.$

11 pages