A height transportation approach to the quantitative Prékopa--Leindler inequality

@misc{machado2026height,
  title = {{A height transportation approach to the quantitative Prékopa--Leindler inequality}},
  author = {João Miguel Machado and João P. G. Ramos},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03295},
  eprint = {2611.03295},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03295},
  note = {CC BY-SA 4.0},
}

TY  - UNPB
TI  - A height transportation approach to the quantitative Prékopa--Leindler inequality
AU  - João Miguel Machado
AU  - João P. G. Ramos
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03295
UR  - https://arxiv.org/abs/2611.03295
AB  - We give a new, simpler proof of sharp stability for the Prékopa--Leindler inequality. In contrast to the main arguments from Figalli, van Hintum and Tiba~\cite{FigalliVanHintumTibaPL}, our proof hinges mostly on the sharp stability theory for the Brunn--Minkowski problem, developed by the same authors \cite{figalli2024BM}, and on the idea of using a height transportation approach in order to match level sets. As a by-product of our techniques, we are able to prove the sharp dependency on τ= min{λ,1−λ} in the Prékopa--Leindler stability constant whenever one of the inputs is logarithmically concave, producing a new, simplified proof of two of the main results from~\cite{FigalliVanHintumTibaPL}. Finally, in the same fashion as Kristály and Pîrvuceanu \cite{kristaly2026quantitative}, we are able to obtain new stability results for the Bakry-Emery log-Sobolev inequality and for the Talagrand inequality. Due to the sharpness of the inputs, these results are optimal in terms of exponents. We moreover show that the L^1 nature of the distance in such results \emph{cannot} in general be improved to any other L^p distance, where p>1.
ER  -