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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.08909 |
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| _version_ | 1866915997527572480 |
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| author | He, Runtai |
| author_facet | He, Runtai |
| contents | We study the discrete graph-metric analogue of Gromov's filling area problem for the cycle graph \(C_n\). An abstract triangulation \(K\) is an isometric filling of \(C_n\) if \(\partial K=C_n\) and the graph distance between any two boundary vertices is not shortened inside the \(1\)-skeleton of \(K\). Let \(D(n;ε)\) denote the minimum number of vertices in a \((1-ε)\)-Lipschitz filling of \(C_n\), and set \[ D^*=\liminf_{ε\to0^+}\liminf_{n\to\infty}\frac{D(n;ε)}{n^2}. \] Previous work gives the general lower bound \(D^*\ge 1/8\), while discretizing the hemisphere gives the upper bound \[ D^*\le \frac{1}{π\sqrt3}. \] In this paper we give an explicit discrete construction which improves the hemispherical upper bound. More precisely, we construct isometric fillings \(K_n\) of \(C_n\) with \[ |V(K_n)|\le \left(\frac16+o(1)\right)n^2, \] and hence \[ D^*\le \frac16<\frac{1}{π\sqrt3}. \] This can directly illustrate the discrete filling area problem is a proper relaxation of Gromov's original filling area problem and cannot be used to settle Gromov's conjecture. The construction is a concentric annular filling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08909 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Upper Bound for Discrete Isometric Filling of Cycles He, Runtai Differential Geometry Combinatorics We study the discrete graph-metric analogue of Gromov's filling area problem for the cycle graph \(C_n\). An abstract triangulation \(K\) is an isometric filling of \(C_n\) if \(\partial K=C_n\) and the graph distance between any two boundary vertices is not shortened inside the \(1\)-skeleton of \(K\). Let \(D(n;ε)\) denote the minimum number of vertices in a \((1-ε)\)-Lipschitz filling of \(C_n\), and set \[ D^*=\liminf_{ε\to0^+}\liminf_{n\to\infty}\frac{D(n;ε)}{n^2}. \] Previous work gives the general lower bound \(D^*\ge 1/8\), while discretizing the hemisphere gives the upper bound \[ D^*\le \frac{1}{π\sqrt3}. \] In this paper we give an explicit discrete construction which improves the hemispherical upper bound. More precisely, we construct isometric fillings \(K_n\) of \(C_n\) with \[ |V(K_n)|\le \left(\frac16+o(1)\right)n^2, \] and hence \[ D^*\le \frac16<\frac{1}{π\sqrt3}. \] This can directly illustrate the discrete filling area problem is a proper relaxation of Gromov's original filling area problem and cannot be used to settle Gromov's conjecture. The construction is a concentric annular filling. |
| title | An Upper Bound for Discrete Isometric Filling of Cycles |
| topic | Differential Geometry Combinatorics |
| url | https://arxiv.org/abs/2605.08909 |