On Busemann--Hausdorff densities of dimension two and of codimension two, with an application to Plateau Problem
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915050547052544 |
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| author | Vasilyev, Ioann |
| author_facet | Vasilyev, Ioann |
| contents | The purpose of this paper is twofold. First, we describe one (presumably) new case, in which Busemann--Hausdorff densities are convex. We apply the corresponding result to prove the existence of minimizing rectifiable chains of codimension two in complex finite dimensional normed vector spaces. Second, we prove that for each $n\geq 4$, there exists an $n$ dimensional normed space in which the corresponding two dimensional Busemann--Hausdroff density is not totally convex. This gives a negative answer to a question posed by H. Busemann and E. Strauss. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_17160 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Busemann--Hausdorff densities of dimension two and of codimension two, with an application to Plateau Problem Vasilyev, Ioann Functional Analysis Metric Geometry 28A75, 49Q15, 49Q20, 28A78, 52A51 The purpose of this paper is twofold. First, we describe one (presumably) new case, in which Busemann--Hausdorff densities are convex. We apply the corresponding result to prove the existence of minimizing rectifiable chains of codimension two in complex finite dimensional normed vector spaces. Second, we prove that for each $n\geq 4$, there exists an $n$ dimensional normed space in which the corresponding two dimensional Busemann--Hausdroff density is not totally convex. This gives a negative answer to a question posed by H. Busemann and E. Strauss. |
| title | On Busemann--Hausdorff densities of dimension two and of codimension two, with an application to Plateau Problem |
| topic | Functional Analysis Metric Geometry 28A75, 49Q15, 49Q20, 28A78, 52A51 |
| url | https://arxiv.org/abs/2203.17160 |