A geometric approach to Poincaré inequality and Minkowski content of separating sets
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arXiv
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| Format: | Preprint |
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2024
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| author | Caputo, Emanuele Cavallucci, Nicola |
| author_facet | Caputo, Emanuele Cavallucci, Nicola |
| contents | The goal of this paper is to continue the study of the relation between the Poincaré inequality and the lower bounds of Minkowski content of separating sets, initiated in our previous work [Caputo, Cavallucci: Poincaré inequality and energy of separating sets, arXiv 2401.02762]. A new shorter proof is provided. An intermediate tool is the study of the lower bound of another geometric quantity, called separating ratio. The main novelty is the description of the relation between the infima of the separating ratio and the Minkowski content of separating sets. We prove a quantitative comparison between the two infima in the local quasigeodesic case and equality in the local geodesic one. No Poincaré assumption is needed to prove it. The main tool employed in the proof is a new function, called the position function, which allows in a certain sense to fibrate a set in boundaries of separating sets. We also extend the proof to measure graphs, where due to the combinatorial nature of the problem, the approach is more intuitive. In the appendix, we revise some classical characterizations of the p-Poincaré inequality, by proving along the way equivalence with a notion of p-pencil that extends naturally the definition for p = 1. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_18327 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A geometric approach to Poincaré inequality and Minkowski content of separating sets Caputo, Emanuele Cavallucci, Nicola Metric Geometry Functional Analysis 30L99, 28A75, 49J52 The goal of this paper is to continue the study of the relation between the Poincaré inequality and the lower bounds of Minkowski content of separating sets, initiated in our previous work [Caputo, Cavallucci: Poincaré inequality and energy of separating sets, arXiv 2401.02762]. A new shorter proof is provided. An intermediate tool is the study of the lower bound of another geometric quantity, called separating ratio. The main novelty is the description of the relation between the infima of the separating ratio and the Minkowski content of separating sets. We prove a quantitative comparison between the two infima in the local quasigeodesic case and equality in the local geodesic one. No Poincaré assumption is needed to prove it. The main tool employed in the proof is a new function, called the position function, which allows in a certain sense to fibrate a set in boundaries of separating sets. We also extend the proof to measure graphs, where due to the combinatorial nature of the problem, the approach is more intuitive. In the appendix, we revise some classical characterizations of the p-Poincaré inequality, by proving along the way equivalence with a notion of p-pencil that extends naturally the definition for p = 1. |
| title | A geometric approach to Poincaré inequality and Minkowski content of separating sets |
| topic | Metric Geometry Functional Analysis 30L99, 28A75, 49J52 |
| url | https://arxiv.org/abs/2402.18327 |