Differentiability of Lipschitz curves in $p$-Banach spaces

The problem of differentiating Lipschitz curves in Banach spaces goes back to Tamarkin in the early 1930s and is one of the origins of the Radon–Nikodým property in classical Banach space theory. In this article we characterize those quasi-Banach spaces $X$ for which every Lipschitz curve $F\to \mathbb{R}\to X$ admits a point of differentiability and prove that this property holds if and only if $X$ is isomorphic to a Banach space with the Radon–Nikodým property, thus substantiating a conjecture of Kalton, communicated personally to the first-named author in 2008. Equivalently, every nonlocally convex quasi-Banach space admits a nowhere differentiable Lipschitz curve. Our result is obtained from a quantitative characterization of local convexity in terms of the infinitesimal oscillation of Lipschitz curves. Motivated by this obstruction, we then investigate positive differentiability phenomena in ${\mathrm{\ell }}_p$, $0<p<1$, and identify natural decoupled-coordinate constructions for which Lipschitz regularity nevertheless implies almost-everywhere differentiability. We also contrast this behavior with metric differentiability for the natural metric and the quasi-metric on ${\mathrm{\ell }}_p$.