An extremal vector-valued $L^{p}$-function taking no extremal vectors as values

Abstract:

We give an example of a nonseparable Banach space $V$ and a function $x$ on [0, 1] with values in the unit sphere of $V$ that is an extreme point of the unit balls of all Bochner ${L^p}$-spaces ${L^p}(\lambda ,V)$, $1 < p \leqslant \infty$, $\lambda$ Lebesgue measure, though none of its values is an extreme point of the unit ball of $V$. This shows that a characterization of the extremal elements in ${L^p}(\lambda ,V)$ for separable $V$, given by J. A. Johnson, does not hold in general.