Stability constants and the homology of quasi-Banach spaces

TL;DR

This paper resolves two longstanding open problems in the theory of quasi-Banach spaces, demonstrating the equivalence of stability constants and homological properties, and providing new insights into their structure.

Contribution

It proves the equivalence of two major open problems related to stability constants and homology in quasi-Banach spaces, advancing the understanding of their structure.

Findings

Existence of cases with finite second Whitney constant but infinite approximation constant.

Existence of Banach spaces with non-zero, Hausdorff Ext groups.

Equivalence of the two main open problems in the field.

Abstract

We affirmatively solve the main problems posed by Laczkovich and Paulin in \emph{Stability constants in linear spaces}, Constructive Approximation 34 (2011) 89--106 (do there exist cases in which the second Whitney constant is finite while the approximation constant is infinite?) and by Cabello and Castillo in \emph{The long homology sequence for quasi-Banach spaces, with applications}, Positivity 8 (2004) 379--394 (do there exist Banach spaces $X,Y$ for which $\Ext(X,Y)$ is Hausdorff and non-zero?). In fact, we show that these two problems are the same.