Lower Bounding the Gromov--Hausdorff distance in Metric Graphs
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909974046703616 |
|---|---|
| author | Adams, Henry Majhi, Sushovan Manin, Fedor Virk, Žiga Zava, Nicolò |
| author_facet | Adams, Henry Majhi, Sushovan Manin, Fedor Virk, Žiga Zava, Nicolò |
| contents | Let $G$ be a finite, connected metric graph and let $X\subseteq G$ be a subset. If $X$ is sufficiently dense in $G$, we show that the Gromov--Hausdorff distance matches the Hausdorff distance, namely $d_\gh(G,X)=d_\h(G,X)$. When the metric graph is the circle $G=S^1$ with circumference $2π$, a recent study established the equality $d_\gh(S^1,X)=d_\h(S^1,X)$ whenever $d_\gh(S^1,X)<\fracπ{6}$. Our results relax this hypothesis to $d_\gh(S^1,X)<\fracπ{3}$, and furthermore, we show that the constant $\fracπ{3}$ is the best possible. We lower bound the Gromov--Hausdorff distance $d_\gh(G,X)$ by the Hausdorff distance $d_\h(G,X)$ via a simple topological obstruction: the existence of a possibly discontinuous function $f\colon G \to X$ with too small distortion contradicts the connectedness of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09182 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower Bounding the Gromov--Hausdorff distance in Metric Graphs Adams, Henry Majhi, Sushovan Manin, Fedor Virk, Žiga Zava, Nicolò Metric Geometry 51F99, 52C25, 54H25 Let $G$ be a finite, connected metric graph and let $X\subseteq G$ be a subset. If $X$ is sufficiently dense in $G$, we show that the Gromov--Hausdorff distance matches the Hausdorff distance, namely $d_\gh(G,X)=d_\h(G,X)$. When the metric graph is the circle $G=S^1$ with circumference $2π$, a recent study established the equality $d_\gh(S^1,X)=d_\h(S^1,X)$ whenever $d_\gh(S^1,X)<\fracπ{6}$. Our results relax this hypothesis to $d_\gh(S^1,X)<\fracπ{3}$, and furthermore, we show that the constant $\fracπ{3}$ is the best possible. We lower bound the Gromov--Hausdorff distance $d_\gh(G,X)$ by the Hausdorff distance $d_\h(G,X)$ via a simple topological obstruction: the existence of a possibly discontinuous function $f\colon G \to X$ with too small distortion contradicts the connectedness of $G$. |
| title | Lower Bounding the Gromov--Hausdorff distance in Metric Graphs |
| topic | Metric Geometry 51F99, 52C25, 54H25 |
| url | https://arxiv.org/abs/2411.09182 |