$k-$smoothness on polyhedral Banach spaces

TL;DR

This paper characterizes the concept of $k$-smoothness in finite-dimensional polyhedral Banach spaces and investigates the $k$-smoothness of certain operators, providing new insights into their geometric properties.

Contribution

It introduces characterizations of $k$-smoothness for elements and operators in specific finite-dimensional Banach spaces, extending existing geometric analysis.

Findings

Characterization of $k$-smoothness on the unit sphere of polyhedral Banach spaces.

Analysis of $k$-smoothness of operators from $ell_{ty}^n$ to two-dimensional Banach spaces.

Characterization of $k$-smoothness of operators between $ell_{ty}^3$ and $ell_{1}^3$.

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 $\ell_{1}^3.$