Improved stability versions of the Prékopa-Leindler inequality
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914962439405568 |
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| author | Figalli, Alessio Ramos, João P. G. |
| author_facet | Figalli, Alessio Ramos, João P. G. |
| contents | We consider the problem of stability for the Prékopa-Leindler inequality. Exploiting properties of the transport map between radially decreasing functions and a suitable functional version of the trace inequality, we obtain a uniform stability exponent for the Prékopa-Leindler inequality.
Our result yields an exponent not only uniform in the dimension but also in the log-concavity parameter $τ= \min(λ,1-λ)$ associated with its respective version of the Prékopa-Leindler inequality. As a further application of our methods, we prove a sharp stability result for log-concave functions in dimension 1, which also extends to a sharp stability result for log-concave radial functions in higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01122 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved stability versions of the Prékopa-Leindler inequality Figalli, Alessio Ramos, João P. G. Functional Analysis Metric Geometry We consider the problem of stability for the Prékopa-Leindler inequality. Exploiting properties of the transport map between radially decreasing functions and a suitable functional version of the trace inequality, we obtain a uniform stability exponent for the Prékopa-Leindler inequality. Our result yields an exponent not only uniform in the dimension but also in the log-concavity parameter $τ= \min(λ,1-λ)$ associated with its respective version of the Prékopa-Leindler inequality. As a further application of our methods, we prove a sharp stability result for log-concave functions in dimension 1, which also extends to a sharp stability result for log-concave radial functions in higher dimensions. |
| title | Improved stability versions of the Prékopa-Leindler inequality |
| topic | Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2410.01122 |