Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910777200345088 |
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| author | Figalli, Alessio van Hintum, Peter Tiba, Marius |
| author_facet | Figalli, Alessio van Hintum, Peter Tiba, Marius |
| contents | The Borell-Brascamp-Lieb inequality is a classical extension of the Prékopa-Leindler inequality, which in turn is a functional counterpart of the Brunn-Minkowski inequality. The stability of these inequalities has received significant attention in recent years. Despite substantial progress in the geometric setting, a sharp quantitative stability result for the Prékopa-Leindler inequality has remained elusive, even in the special case of log-concave functions. In this work, we provide a unified and definitive stability framework for these foundational inequalities. By establishing the optimal quantitative stability for the Borell-Brascamp-Lieb inequality in full generality, we resolve the conjectured sharp stability for the Prékopa-Leindler inequality as a particular case. Our approach builds on the recent sharp stability results for the Brunn-Minkowski inequality obtained by the authors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities Figalli, Alessio van Hintum, Peter Tiba, Marius Functional Analysis Analysis of PDEs Combinatorics Metric Geometry Probability 52A40, 49Q20, 49Q22, 52A27, 26D15 The Borell-Brascamp-Lieb inequality is a classical extension of the Prékopa-Leindler inequality, which in turn is a functional counterpart of the Brunn-Minkowski inequality. The stability of these inequalities has received significant attention in recent years. Despite substantial progress in the geometric setting, a sharp quantitative stability result for the Prékopa-Leindler inequality has remained elusive, even in the special case of log-concave functions. In this work, we provide a unified and definitive stability framework for these foundational inequalities. By establishing the optimal quantitative stability for the Borell-Brascamp-Lieb inequality in full generality, we resolve the conjectured sharp stability for the Prékopa-Leindler inequality as a particular case. Our approach builds on the recent sharp stability results for the Brunn-Minkowski inequality obtained by the authors. |
| title | Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities |
| topic | Functional Analysis Analysis of PDEs Combinatorics Metric Geometry Probability 52A40, 49Q20, 49Q22, 52A27, 26D15 |
| url | https://arxiv.org/abs/2501.04656 |