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

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~{FigalliVanHintumTibaPL}, our proof hinges mostly on the sharp stability theory for the Brunn--Minkowski problem, developed by the same authors {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~{FigalliVanHintumTibaPL}. Finally, in the same fashion as Kristály and Pîrvuceanu {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 L¹ nature of the distance in such results cannot in general be improved to any other Lᵖ distance, where $p>1.$