The Hausdorff distance and metrics on toric singularity types
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arXiv
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| Autores principales: | , , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909980429385728 |
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| author | Aitokhuehi, Ayo Braiman, Benjamin Cutler, David Owen Horace Darvas, Tamás Deaton, Robert Gupta, Prakhar Horsley, Jude Pidaparthy, Vasanth Tang, Jen |
| author_facet | Aitokhuehi, Ayo Braiman, Benjamin Cutler, David Owen Horace Darvas, Tamás Deaton, Robert Gupta, Prakhar Horsley, Jude Pidaparthy, Vasanth Tang, Jen |
| contents | Given a compact Kähler manifold $(X,ω)$, due to the work of Darvas-Di Nezza-Lu, the space of singularity types of $ω$-psh functions admits a natural pseudo-metric $d_\mathcal S$ that is complete in the presence of positive mass. When restricted to model singularity types, this pseudo-metric is a bona fide metric. In case of the projective space, there is a known one-to-one correspondence between toric model singularity types and convex bodies inside the unit simplex. Hence in this case it is natural to compare the $d_\mathcal S$ metric to the classical Hausdorff metric. We provide precise Hölder bounds, showing that their induced topologies are the same. More generally, we introduce a quasi-metric $d_G$ on the space of compact convex sets inside an arbitrary convex body $G$, with $d_\mathcal S = d_G$ in case $G$ is the unit simplex. We prove optimal Hölder bounds comparing $d_G$ with the Hausdorff metric. Our analysis shows that the Hölder exponents differ depending on the geometry of $G$, with the worst exponents in case $G$ is a polytope, and the best in case $G$ has $C^2$ boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11246 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Hausdorff distance and metrics on toric singularity types Aitokhuehi, Ayo Braiman, Benjamin Cutler, David Owen Horace Darvas, Tamás Deaton, Robert Gupta, Prakhar Horsley, Jude Pidaparthy, Vasanth Tang, Jen Complex Variables Analysis of PDEs Given a compact Kähler manifold $(X,ω)$, due to the work of Darvas-Di Nezza-Lu, the space of singularity types of $ω$-psh functions admits a natural pseudo-metric $d_\mathcal S$ that is complete in the presence of positive mass. When restricted to model singularity types, this pseudo-metric is a bona fide metric. In case of the projective space, there is a known one-to-one correspondence between toric model singularity types and convex bodies inside the unit simplex. Hence in this case it is natural to compare the $d_\mathcal S$ metric to the classical Hausdorff metric. We provide precise Hölder bounds, showing that their induced topologies are the same. More generally, we introduce a quasi-metric $d_G$ on the space of compact convex sets inside an arbitrary convex body $G$, with $d_\mathcal S = d_G$ in case $G$ is the unit simplex. We prove optimal Hölder bounds comparing $d_G$ with the Hausdorff metric. Our analysis shows that the Hölder exponents differ depending on the geometry of $G$, with the worst exponents in case $G$ is a polytope, and the best in case $G$ has $C^2$ boundary. |
| title | The Hausdorff distance and metrics on toric singularity types |
| topic | Complex Variables Analysis of PDEs |
| url | https://arxiv.org/abs/2411.11246 |