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Results tagged with mg.metric-geometry
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user 11142
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
2
votes
Volume of a region given by a Constraint Satisfaction Problem
It all depends on how small $n$ is compared to $m,$ but in general, this is very hard, and the cost of the oracle is the least of your problems: the number of such calls grows exponentially in $n.$ Fo …
- 96.4k
3
votes
Isoperimetric inequality via Crofton's formula
See the below beautiful note of Chris Croke's.
Christopher B. Croke, MR 2361884 A synthetic characterization of the hemisphere, Proc. Amer. Math. Soc. 136 (2008), no. 3, 1083--1086 (electronic).]
- 96.4k
11
votes
Accepted
About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$
This question (in a much more general form) is answered in this preprint by Marichal and Mosinghoff. They point out that the answer to your question actually goes back to Polya's PhD thesis.
- 96.4k
2
votes
Accepted
Is there a general formula for calculating the volume of elliptical simplex on the surface o...
See these notes by J. G. Heckman (he focuses on the hyperbolic case, but the spherical case is essentially identical).
- 96.4k
0
votes
Accepted
Surface integral approximation
This is really a question about Minkowski content. If you look at the cited article (under properties), this seems to indicate that under your Lipschitz condition, the answer to your question is YES ( …
- 96.4k
7
votes
Accepted
How to partition a quadrilateral into a finite number of equal-area triangles
See this Wikipedia article.
- 96.4k
5
votes
Accepted
Euclidean inside Hyperbolic
I believe there is no good model of $\mathbb{E}^2$ in $\mathbb{H}^2.$ However, there is an excellent model in $\mathbb{H}^3:$ any horosphere will work.
Also This is not particularly interesting, but …
- 96.4k
3
votes
A question concerning a well known "law" about triangles.
This is a duplicate of this question, which has good answers.
- 96.4k
2
votes
Escher, Conway, Kali, etc.
Can't speak for the Conway -> generators, but for drawing, there is this
http://www.plunk.org/~hatch/HyperbolicApplet/
I am not sure why "rummaging" was necessary for the D. Huson program, since ther …
- 96.4k
2
votes
Volume ratio of $\ell_1$ balls and $\ell_1$ surfaces
I am a little confused. $S_{d-1}$ is a disjoint union of $2^d$ isometric regular simplices (one of them is the set of $x_1, \dotsc, x_d,$ with $x_i \geq 0,$ and $\sum x_i = 1.$ The volume of such a si …
- 96.4k
4
votes
Curvature of a finite metric space
These notes by John Lott (covering some joint work with Villani) do it for length spaces, which finite metric spaces never are, but if you join the points by edges whose lengths are the distances (so …
- 96.4k
4
votes
3
votes
Accepted
Bound on maximum distance between points on a unit N-Sphere
Call the quantity in question $D(M, N),$ and let the volume of the spherical cap of dimension $N$ and radius $r$ $V(N, r),$normalized so that the volume of the whole sphere is $1.$ Since the caps of r …
- 96.4k
1
vote
Fixed points of finite order isometries of metric spaces
If $X$ is a CAT(0) space, this is true (the standard reference is Bridson-Haefliger, corr. 2.8, though the result precedes them by several decades). I believe the argument also works for $p$-uniformly …
- 96.4k
3
votes
conjugacy between geodesic flows on 2-tori
Yes, this is theorem C in:
Croke, Christopher B.(1-PA)
Rigidity for surfaces of nonpositive curvature.
Comment. Math. Helv. 65 (1990), no. 1, 150–169.
- 96.4k