Generalized and Convex Poincaré Inequalities via Transport-Divergence Inequalities
Abstract
We investigate the relationship between generalized (q ′ , p ′ )-Poincaré inequalities and transport-divergence inequalities involving χ q -type divergences. Extending the classical correspondence between the (2, 2)-Poincaré inequality and quadratic transport-χ 2 inequalities, we prove that a (q ′ , p ′ )-Poincaré inequality is equivalent to a transportdivergence inequality for conjugates (p, p ′ ) and (q, q ′ ). We then study the convex counterpart of these inequalities. Using weak barycentric optimal transport and convex order, we establish an equivalence between convex (q ′ , p ′ )-Poincaré inequalities and weak transport-divergence inequalities involving a projected χ q divergence. We further characterize the minimizers arising in this projection problem through a geometric condition on the relative density. In dimension one, we analyze the geometric consequences of (q ′ , p ′ )-Poincaré inequalities by identifying sharp conditions on the behavior of the density both at infinity and near isolated zeros. We show that when q ′ > p ′ , the concentration behavior becomes polynomial rather than exponential at infinity. Finally, we interpret these inequalities through the lens of optimization in Wasserstein space. In particular, the equivalence between transport-divergence and Poincaré inequalities appears as an analogue of the equivalence between quadratic growth and Polyak-Lojasiewicz inequalities in finite-dimensional optimization.
Related
- Novel Unified Variants, Properties, and Applications of Ostrowski, Jensen, and Hermite–Hadamard Inequalities for Generalized (η, p, h)-Convex Stochastic Processes
- Rellich-Poincaré inequalities and improved Rellich inequalities
- Caffarelli-Kohn-Nirenberg and Weighted Gaussian Poincaré Inequalities: a complete characterization of sharp L2 stability and Lp extensions
- Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations
- Noncommutative Lp-Poincaré inequalities
BibTeX
@misc{gouic2026generalized,
title = {{Generalized and Convex Poincaré Inequalities via Transport-Divergence Inequalities}},
author = {Thibaut Le Gouic and Magali Tournus},
howpublished = {HAL},
year = {2026},
month = jul,
url = {https://hal.science/hal-05780607v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalized and Convex Poincaré Inequalities via Transport-Divergence Inequalities
AU - Thibaut Le Gouic
AU - Magali Tournus
PY - 2026
DA - 2026/07/08
PB - HAL
UR - https://hal.science/hal-05780607v1
AB - We investigate the relationship between generalized (q ′ , p ′ )-Poincaré inequalities and transport-divergence inequalities involving χ q -type divergences. Extending the classical correspondence between the (2, 2)-Poincaré inequality and quadratic transport-χ 2 inequalities, we prove that a (q ′ , p ′ )-Poincaré inequality is equivalent to a transportdivergence inequality for conjugates (p, p ′ ) and (q, q ′ ). We then study the convex counterpart of these inequalities. Using weak barycentric optimal transport and convex order, we establish an equivalence between convex (q ′ , p ′ )-Poincaré inequalities and weak transport-divergence inequalities involving a projected χ q divergence. We further characterize the minimizers arising in this projection problem through a geometric condition on the relative density. In dimension one, we analyze the geometric consequences of (q ′ , p ′ )-Poincaré inequalities by identifying sharp conditions on the behavior of the density both at infinity and near isolated zeros. We show that when q ′ > p ′ , the concentration behavior becomes polynomial rather than exponential at infinity. Finally, we interpret these inequalities through the lens of optimization in Wasserstein space. In particular, the equivalence between transport-divergence and Poincaré inequalities appears as an analogue of the equivalence between quadratic growth and Polyak-Lojasiewicz inequalities in finite-dimensional optimization.
ER -