Power-law asymptotics of fractional $L^p$ polar projection bodies
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918278287327232 |
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| author | Lê, Trí Minh |
| author_facet | Lê, Trí Minh |
| contents | The notion of $s$--fractional $L^p$ polar projection bodies, recently introduced by Haddad and Ludwig (Math.\ Ann.\ \textbf{388}:1091--1115, 2024), provides a bridge between fractional Sobolev theory and convex geometry. In this manuscript, we study the limit of their Minkowski gauges under two natural asymptotic regimes: \\ \hspace*{3em} (a) first sending $p \to \infty$ and then $s \to 1^-$; \\ \hspace*{3em} (b) first sending $s \to 1^-$ and then $p \to \infty$. \\ Our main result shows that these two limiting processes commute. As a consequence, we derive precise asymptotic behavior for the associated volumes and dual mixed volumes, thereby linking the (fractional) $L^p$ polar projection bodies to the newly introduced (fractional) $L^\infty$ polar projection bodies. These results further yield new geometric inequalities, including endpoint Lipschitz/Hölder isoperimetric--type and variants of Pólya--Szegő inequalities in the $L^\infty$ setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05153 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Power-law asymptotics of fractional $L^p$ polar projection bodies Lê, Trí Minh Metric Geometry 46E35, 52A40 (Primary) 35R11, 26D10 (Secondary) The notion of $s$--fractional $L^p$ polar projection bodies, recently introduced by Haddad and Ludwig (Math.\ Ann.\ \textbf{388}:1091--1115, 2024), provides a bridge between fractional Sobolev theory and convex geometry. In this manuscript, we study the limit of their Minkowski gauges under two natural asymptotic regimes: \\ \hspace*{3em} (a) first sending $p \to \infty$ and then $s \to 1^-$; \\ \hspace*{3em} (b) first sending $s \to 1^-$ and then $p \to \infty$. \\ Our main result shows that these two limiting processes commute. As a consequence, we derive precise asymptotic behavior for the associated volumes and dual mixed volumes, thereby linking the (fractional) $L^p$ polar projection bodies to the newly introduced (fractional) $L^\infty$ polar projection bodies. These results further yield new geometric inequalities, including endpoint Lipschitz/Hölder isoperimetric--type and variants of Pólya--Szegő inequalities in the $L^\infty$ setting. |
| title | Power-law asymptotics of fractional $L^p$ polar projection bodies |
| topic | Metric Geometry 46E35, 52A40 (Primary) 35R11, 26D10 (Secondary) |
| url | https://arxiv.org/abs/2601.05153 |