Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Delta-points and their implications for the geometry of Banach spaces (2303.00511v2)

Published 1 Mar 2023 in math.FA

Abstract: We show that the Lipschitz-free space with the Radon--Nikod\'{y}m property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to $\ell_1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of $\ell_2$, with a $\Delta$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $\Delta$-point. Next, we establish powerful relations between existence of $\Delta$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $\Delta$-points for the asymptotic geometry of Banach spaces. In addition, we prove that if $X$ is a Banach space with a shrinking $k$-unconditional basis with $k < 2$, or if $X$ is a Hahn--Banach smooth space with a dual satisfying the Kadets--Klee property, then $X$ and its dual $X*$ fail to contain $\Delta$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $\Delta$-points. Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $\Delta$-points, and we solve an open problem about representation of finitely supported $\Delta$-points in Lipschitz-free spaces.

Summary

We haven't generated a summary for this paper yet.