We characterize k-smoothness of an element on the unit sphere of a finitedimensional polyhedral Banach space. Then we study k-smoothness of an operatorwhere Y is a two-dimensional Banach space with the additional condition that T attains its norm at each extreme point of B ℓ n ∞ . We also characterize k-smoothness of an operator from ℓ 3 ∞ to ℓ 3 1 .
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