Convexity and smoothness of Banach spaces with numerical index one

TL;DR

This paper investigates the geometric properties of Banach spaces with numerical index one, demonstrating that such spaces are highly non-convex and non-smooth unless they are one-dimensional, and explores related properties like lushness.

Contribution

It establishes that Banach spaces with numerical index one lack good convexity or smoothness properties unless one-dimensional, and connects lushness with the absence of strict convexity and smoothness.

Findings

Banach spaces with numerical index one are not convex or smooth unless one-dimensional.

Lush spaces are neither strictly convex nor smooth unless one-dimensional.

Duals of lush infinite-dimensional spaces contain a copy of ℓ₁.

Abstract

We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~1). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $\ell_1$.