On Hahn-Banach smoothness of $L_1$-preduals and related $w^*-w$ point of continuity of unit balls of dual spaces

This article aims to examine the Hahn-Banach smoothness of Banach spaces and its connections to various geometrical aspects. We examine the circumstances that allow linear functionals to have unique norm-preserving extensions, with particular attention to the behavior of these properties in $L_1$-preduals and in spaces of affine continuous functions. Banach spaces which are $L_1$-preduals and also Hahn-Banach smooth are completely characterized. It is demonstrated that if $X$ is an $M$-embedded space then $X^*$ admits a predual which is not weakly Hahn-Banach smooth. It is derived that, when $S$ is a compact convex set where each point in $ext(S)$ is a limit point of $ext(S)$ and also represents a split face, no subspace of $A(S)$ retains the property-$(wU)$ in $A(S)^{**}$. Furthermore, when $X=C_0(L)$, in the context of a locally compact Hausdorff space $L$, the continuity of the identity mapping $I:(B_{X^*},w^*)\to (B_{X^*},w)$ in $ext (B_{X^*})$ significantly influences the subspaces of $X$ that have unique extension property in $X^{**}$. Collectively, this study provides structural characterizations of specialized geometric property, so called Hahn-Banach smoothness, and offers solutions to some natural problems enlisted at the beginning that involve spaces that are $L_1$-preduals and also spaces that are $M$-embedded.