A Banach space $X$ is said to be Hahn-Banach smooth if every linear functional on $X$ has a unique norm-preserving extension over $X^{**}$. Weak Hahn-Banach smoothness is what if the above condition holds only for the norm attaining functionals on $X$. A similar property was defined for a subspace of a Banach space and referred to as property (U) by RR Phelps (see MR0113125(22#3964)). A weakening of this is known as property (wU)(see MR0733942(86b:46027)). By taking suitable renorming on a Banach space it is possible to construct a subspace which has property (wU) but not have property (U). My question is the following.
What is an example of a Banach space which is weakly Hahn-Banach smooth but not Hahn-Banach smooth?
