Isometries without linear shadows: failure of Figiel's theorem beyond local convexity
Abstract
A classical theorem of Figiel asserts that every isometric embedding φ: X→ Y between real Banach spaces, with φ(0)=0, admits a linear contractive left inverse on the closed linear span of φ(X). We show that this phenomenon breaks down completely once local convexity is lost. More precisely, for every 0<p<1 and every nonzero real Banach space X, we construct a p-Banach space Y and an isometric embedding φ: X→ Y such that φ(0)=0 and the closed linear span of φ(X) is the entire space Y, but there is no linear map, bounded or not, on the linear span of φ(X) which is a left inverse of φ. The construction is carried out using the p-Banach counterpart of Lipschitz-free spaces. In fact, Y may be chosen isomorphic to the Lipschitz-free p-space over X. The main ingredient is a quantitative estimate controlling the distance to elementary molecules in Lipschitz-free p-spaces, a phenomenon specific to the range 0<p<1. As a further consequence, we show that a real Banach space has the isometric p-lifting property if and only if its dimension is at most one.
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BibTeX
@misc{albiac2026isometries,
title = {{Isometries without linear shadows: failure of Figiel's theorem beyond local convexity}},
author = {Fernando Albiac and José L. Ansorena and Marek Cúth},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07448},
eprint = {2610.07448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07448},
}
RIS
TY - UNPB
TI - Isometries without linear shadows: failure of Figiel's theorem beyond local convexity
AU - Fernando Albiac
AU - José L. Ansorena
AU - Marek Cúth
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07448
UR - https://arxiv.org/abs/2610.07448
AB - A classical theorem of Figiel asserts that every isometric embedding φ: X→ Y between real Banach spaces, with φ(0)=0, admits a linear contractive left inverse on the closed linear span of φ(X). We show that this phenomenon breaks down completely once local convexity is lost. More precisely, for every 0<p<1 and every nonzero real Banach space X, we construct a p-Banach space Y and an isometric embedding φ: X→ Y such that φ(0)=0 and the closed linear span of φ(X) is the entire space Y, but there is no linear map, bounded or not, on the linear span of φ(X) which is a left inverse of φ. The construction is carried out using the p-Banach counterpart of Lipschitz-free spaces. In fact, Y may be chosen isomorphic to the Lipschitz-free p-space over X. The main ingredient is a quantitative estimate controlling the distance to elementary molecules in Lipschitz-free p-spaces, a phenomenon specific to the range 0<p<1. As a further consequence, we show that a real Banach space has the isometric p-lifting property if and only if its dimension is at most one.
ER -