Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
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Bibliographic record
Abstract
We investigate Prékopa-Leindler type inequalities on a Riemannian manifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> equipped with a measure with density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:msup> </mml:math> where the potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> and the Ricci curvature satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo form="prefix">Hess</mml:mo> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>V</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mo form="prefix">Ric</mml:mo> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>≥</mml:mo> <mml:mi>λ</mml:mi> <mml:mspace width="0.166667em"/> <mml:mi>I</mml:mi> </mml:mrow> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> , with some <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℝ</mml:mi> </mml:mrow> </mml:math> . As in our earlier work [14], the argument uses optimal mass transport on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> , but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the Bakry-Emery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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