Functional inequalities along Wasserstein geodesics
Abstract
We study functional inequalities along Wasserstein geodesics. If µ 0 and µ 1 are respectively κ 0 -and κ 1 -strongly log-concave probability measures on R n , we prove that their quadratic Wasserstein geodesic satisfies
for all probability measures ν on R n . The coefficient is sharp. By linearization, this recovers the Poincaré estimate of Han and Zhu [15]. On the real line, we prove convexity of the square roots of the optimal T 1 and T 2 constants along monotone interpolation between arbitrary probability measures. The argument applies to more general transport entropy inequalities. We also establish convexity of the rescaled L p Poincaré constants for every finite p ≥ 1, including the square root of the Poincaré constant and the inverse Cheeger constant. Finally, we construct a planar Wasserstein geodesic whose endpoints satisfy T 2 and all finite-p L p -Poincaré inequalities, whereas every interior interpolant fails these inequalities.
Related
- Sharp Poincaré interpolation along Wasserstein geodesics
- Spatial degeneracy along weak geodesics in the space of Kähler potentials
- Quasi-convexity of energy functions along Teichmüller geodesics
- Branched Covers and Fiberwise Degeneration along Weak Kähler Geodesics
- Transport along Local Annealing Paths: Wasserstein Geometry and Multimodal Sampling
BibTeX
@misc{gozlan2026functional,
title = {{Functional inequalities along Wasserstein geodesics}},
author = {Nathael Gozlan and Hugo Malamut and Irène Waldspurger},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39252},
eprint = {2609.39252},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05769896v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Functional inequalities along Wasserstein geodesics
AU - Nathael Gozlan
AU - Hugo Malamut
AU - Irène Waldspurger
PY - 2026
DA - 2026/09/29
PB - HAL
DO - 10.48550/arxiv.2609.39252
UR - https://hal.science/hal-05769896v1
AB - We study functional inequalities along Wasserstein geodesics. If µ 0 and µ 1 are respectively κ 0 -and κ 1 -strongly log-concave probability measures on R n , we prove that their quadratic Wasserstein geodesic satisfies for all probability measures ν on R n . The coefficient is sharp. By linearization, this recovers the Poincaré estimate of Han and Zhu [15]. On the real line, we prove convexity of the square roots of the optimal T 1 and T 2 constants along monotone interpolation between arbitrary probability measures. The argument applies to more general transport entropy inequalities. We also establish convexity of the rescaled L p Poincaré constants for every finite p ≥ 1, including the square root of the Poincaré constant and the inverse Cheeger constant. Finally, we construct a planar Wasserstein geodesic whose endpoints satisfy T 2 and all finite-p L p -Poincaré inequalities, whereas every interior interpolant fails these inequalities.
ER -