The Legendre transform, the Laplace transform and valuations
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911507099418624 |
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| author | Li, Jin |
| author_facet | Li, Jin |
| contents | We first prove that the Legendre transform is the only continuous and $\mathrm{SL}(n)$ contravariant valuation that behaves as a conjugation of two important translations on super-coercive, lower semi-continuous, and convex functions. Then we turn to a similar setting on log-concave functions and find characterizations of not merely the duality transform but also the Laplace transform on log-concave functions. With the notion of dual valuation, we also obtain characterizations of the identity transform on finite convex functions and positive log-concave functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07022 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Legendre transform, the Laplace transform and valuations Li, Jin Metric Geometry Functional Analysis 52B45, 52A41, 52A20, 26B25, 44A10, 49N15 We first prove that the Legendre transform is the only continuous and $\mathrm{SL}(n)$ contravariant valuation that behaves as a conjugation of two important translations on super-coercive, lower semi-continuous, and convex functions. Then we turn to a similar setting on log-concave functions and find characterizations of not merely the duality transform but also the Laplace transform on log-concave functions. With the notion of dual valuation, we also obtain characterizations of the identity transform on finite convex functions and positive log-concave functions. |
| title | The Legendre transform, the Laplace transform and valuations |
| topic | Metric Geometry Functional Analysis 52B45, 52A41, 52A20, 26B25, 44A10, 49N15 |
| url | https://arxiv.org/abs/2308.07022 |