Journal of the
Korean Mathematical Society JKMS

We prove that a Banach space that is convex-transitive and such that for some element $u$ in the unit sphere, and for every subspace $M$ containing $u$, it happens that the subset of norm attaining functionals on $M$ is second Baire category in $M^\ast$ is, in fact, almost-transitive and superreflexive. We also obtain a characterization of finite-dimensional spaces in terms of functions that attain their supremum: a Banach space is finite-dimensional if, for every equivalent norm, every rank-one operator attains its numerical radius. Finally, we describe the subset of norm attaining functionals on a space isomorphic to $\ell_1$, where the norm is the restriction of a Luxembourg norm on $L_1$. In fact, the subset of norm attaining functionals for this norm coincides with the subset of norm attaining functionals for the usual norm.

Keywords: reflexive Banach spaces, norm attaining functionals, convex-transitive Banach spaces, almost-transitive Banach spaces, numerical radius attaining operators, Luxembourg norm, smooth norm

MSC numbers: 46B10, 46B03, 46B45, 47A12