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Stability constants and the homology of quasi-Banach spaces

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.