Skewness in Banach spaces

Abstract:

Let $E$ be a Banach space. One often wants to measure how far $E$ is from being a Hilbert space. In this paper we define the skewness $s(E)$ of a Banach space $E$, $0 \leqslant s(E) \leqslant 2$, which describes the asymmetry of the norm. We show that $s(E) = s({E^{\ast }})$ for all Banach spaces $E$. Further, $s(E) = 0$ if and only if $E$ is a (real) Hilbert space and $s(E) = 2$ if and only if $E$ is quadrate, so $s(E) < 2$ implies $E$ is reflexive. We discuss the computation of $s({L^p})$ and describe its asymptotic behavior near $p = 1,2$ and $\infty$. Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.