On Hahn-Banach smoothness and related properties in Banach spaces

In this paper, we study several variants of Hahn-Banach smoothness, viz., property-$(SU)$/$(HB)$/$(wU)$, where property-$(SU)$ and property-$(HB)$ are stronger notions and property-$(wU)$ is a weaker notion of Hahn-Banach smoothness. We characterize property-$(wU)$ and property-$(HB)$. It is observed that $L_1(\mu)$ has property-$(wU)$ in $L_1(\mu,(\mathbb{R}^2,\|.\|_2))$ but it does not have property-$(U)$ in $L_1(\mu,(\mathbb{R}^2,\|.\|_2))$ for a non-atomic measure $\mu$. We derive a sufficient condition when property-$(wU)$ is equivalent to property-$(U)$ of a subspace. It is observed that these properties are separably determined. Finally, finite-dimensional and finite co-dimensional subspaces of $c_0$, $\ell_p$ ($1\leq p<\infty$) having these properties are characterized.