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Results tagged with mg.metric-geometry
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user 4558
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
9
votes
1
answer
479
views
What does convergence in distribution "in the Gromov–Hausdorff" sense mean?
I am trying to understand this survey article by Le Gall on Brownian geometry, especially the statement of Theorem 1.
The basic statement of the theorem is
$$(m_n,d_n) \to (m_{\infty}, d_{\infty})$$
…
- 7,857
8
votes
Is there a dense subset of the real plane with all pairwise distances rational?
Victor Klee and Stan Wagon write about this and other fun problems in their book:
Old and New Unsolved Problems in Plane Geometry and Number Theory
- 7,857
8
votes
1
answer
700
views
Upper bound for tetrahedron packing?
There have been several recent advances on packing regular tetrahedra in $\mathbb{R}^3$. All the results I've seen have been lower bounds -- first John Conway and Sal Torquato showed that there exist …
- 7,857
3
votes
Applications of the notion of of Gromov-Hausdorff distance
Facundo Memoli applied Gromov-Hausdorff distance to shape matching in his Ph.D. thesis.
http://math.stanford.edu/~memoli/research.html (Wayback Machine, new website)
8
votes
1
answer
309
views
Longest induced cycles in random geometric graphs near criticality
We make a random geometric graph $X(n;r)$ as follows. Choose $n$ points uniformly, independently, in the unit square $[0,1]^2$, for vertices, and then connect a pair of vertices $\{ p,q \}$ by an edge …
- 7,857
5
votes
1
answer
295
views
Updated bounds or references for an old Erdős problem –– coloring the plane with multiple fo...
Define a graph $G_1$ where the vertices of $G_1$ are the points of the plane $\mathbb{R}^2$, and a pair of vertices $p, q$ is connected by an edge if and only if the Euclidean distance $d(p,q) =1$. Th …
- 7,857
11
votes
2
answers
3k
views
Algorithm for embedding a graph with metric constraints
Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, a poistive integer $d$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide …
- 7,857
4
votes
Accepted
Average degree of contact graph for balls in a box
Torquato and Stillinger have a recent survey article that discusses some questions like this: Jammed hard-particle packings: From Kepler to Bernal and beyond
They are particularly interested in rand …
- 7,857
33
votes
Open problems in Euclidean geometry?
The Unit Distance Problem asks:
For a set of $n$ points in the plane,
what is the maximal number $g(n)$ of
unit distances realized among the
${n \choose 2}$ pairs?
A properly scaled squar …