Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $k-$smoothness on polyhedral Banach spaces   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
$\ell_{1}^3.$

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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: stat