Differentiability of Lipschitz curves in p-Banach spaces
Abstract
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→ℝ→ 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 ℓp, 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 ℓp.
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BibTeX
@misc{albiac2026differentiability,
title = {{Differentiability of Lipschitz curves in $p$-Banach spaces}},
author = {Fernando Albiac and José L. Ansorena and Pietro Wald},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26992},
eprint = {2609.26992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26992},
}
RIS
TY - UNPB
TI - Differentiability of Lipschitz curves in p-Banach spaces
AU - Fernando Albiac
AU - José L. Ansorena
AU - Pietro Wald
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26992
UR - https://arxiv.org/abs/2609.26992
AB - 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→ℝ→ 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 ℓ_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 ℓ_p.
ER -