Dense single extensions points in Hahn-Banach theorem

Abstract

Let $X$ be a real separable Banach space. It is shown that the set of points $x\in X\backslash\left\{  0\right\}  $ for which there exists more than one linear and continuous functional $x^{\ast}\in X^{\ast}$ that satisfies $\left\langle x^{\ast},x\right\rangle =\left\Vert x\right\Vert $ and $\left\Vert x^{\ast}\right\Vert =1$\ has no interior. If $\dim X<\infty$, then the set has Lebesgue measure zero.