The Monge-Kantorovich tangent bundle to a measure
Abstract
We study the tangent bundle Tμ to a Radon measure μ in Euclidean space, introduced by Bouchitté, Champion and Jimenez (2005). Its construction, inspired by Monge-Kantorovich optimal transport theory, involves the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures. Precisely, a velocity field σ is μ-tangent when the divergence of σμ lies in the Arens-Eells space. The tangent bundle Tμ defined μ-almost everywhere provides a local projection that allows to construct a μ-tangential gradient operator on Lipschitz functions that is weakly continuous and enjoys integration by parts. In this paper, we introduce a new quantitative estimate involving the tangential and normal components of a given velocity vector field σ∈ L1μ(ℝd). Specifically, we show that the tangential condition σ∈ Tμ holding μ a.e. is equivalent to each of the following two conditions: the convergence in the Arens-Eells space of h−1 ((id+h σ)# μ− μ) as h → 0, and the differentiability of Lipschitz functions along σ. Moreover, given a field of non-tangent directions, we construct a Lipschitz function that is not differentiable on a large set, confirming that Tμ agrees with the decomposability bundle of Alberti and Marchese. We finally relate Tμ to the tangent measures of Preiss and survey further properties of the tangential differential calculus.
Related
- Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance
- A Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows via Monge-Growth Pairs
- Properly embedded Legendrian surfaces in the tangent sphere bundle of hyperbolic space
- Deformations of the logarithmic tangent bundle of smooth projective surfaces
- η− Hyperbolic Ricci Solitons on the Unit Tangent Bundle of the Hyperbolic Strip
BibTeX
@misc{arroyorabasa2026mongekantorovich,
title = {{The Monge-Kantorovich tangent bundle to a measure}},
author = {Adolfo Arroyo-Rabasa and Averil Aussedat and Guy Bouchitté},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29401},
eprint = {2609.29401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29401},
}
RIS
TY - UNPB
TI - The Monge-Kantorovich tangent bundle to a measure
AU - Adolfo Arroyo-Rabasa
AU - Averil Aussedat
AU - Guy Bouchitté
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29401
UR - https://arxiv.org/abs/2609.29401
AB - We study the tangent bundle T_μ to a Radon measure μ in Euclidean space, introduced by Bouchitté, Champion and Jimenez (2005). Its construction, inspired by Monge-Kantorovich optimal transport theory, involves the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures. Precisely, a velocity field σ is μ-tangent when the divergence of σμ lies in the Arens-Eells space. The tangent bundle T_μ defined μ-almost everywhere provides a local projection that allows to construct a μ-tangential gradient operator on Lipschitz functions that is weakly continuous and enjoys integration by parts. In this paper, we introduce a new quantitative estimate involving the tangential and normal components of a given velocity vector field σ∈ L^1_μ(ℝ^d). Specifically, we show that the tangential condition σ∈ T_μ holding μ a.e. is equivalent to each of the following two conditions: the convergence in the Arens-Eells space of h^−1 ((id+h σ)_# μ− μ) as h → 0, and the differentiability of Lipschitz functions along σ. Moreover, given a field of non-tangent directions, we construct a Lipschitz function that is not differentiable on a large set, confirming that T_μ agrees with the decomposability bundle of Alberti and Marchese. We finally relate T_μ to the tangent measures of Preiss and survey further properties of the tangential differential calculus.
ER -