The study of geometry of manifolds without appealing to differential calculus. It includes studies of length spaces, Alexandrov spaces, and CAT(k) spaces. The techniques are often applicable to Riemannian/Finsler geometry (where differential calculus is used) and geometric group theory. For questions about plain-old metric spaces, please use (metric-spaces) instead.
Questions tagged [metric-geometry]
282 questions
6
votes
2 answers
Length of a curve defined by a convex function
Let $f \colon [0,1] \to [0,1]$ be a function of class $C^1$ such that $f(0)=f(1)=1$ and $f'$ is nondecreasing, i.e., $f$ is convex. Show that the length of the curve defined by the graph of $f$ is less than or equal to 3.
nom
- 579
5
votes
1 answer
Geodesics meeting with angle $0$ in ${\rm CAT}(0)$ space
Consider two distinct geodesics $\gamma_1$ and $\gamma_2$ in a CAT($0$) space, issued from the same base point.
A trivial example where we have $\angle(\gamma_1, \gamma_2)=0$ is when
$\gamma_1(t) = \gamma_2(t)$ for $t$ smaller than some $\varepsilon…
FMB
- 332
5
votes
0 answers
How to find the "area" of a box (hyperrectangle) projected onto a ball (hypersphere)?
What is the "area" of the general $n-1$ - box (hyperrectangle) projected onto the surface of the general $n$ - ball (hypersphere)?
Is there a known closed-form solution?
I'm just curious. Thanks!
Tdonut
- 3,948
5
votes
1 answer
why are CAT(0) spaces contractible?
In the book of Bridson and Haefliger it is said that 'it follows easily' from what they proved before. Does anyone know of a rigorous proof that CAT(0) spaces are contractible?
mathreader
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5
votes
1 answer
general formula for volume of a simplex?
I am looking for a general formula to calculate the volume of a euclidean simplex in any number of dimensions. On Wikipedia I found that a formula similar to Heron's formula can be applied to tetrahedrons as well as triangles, can this perhaps be…
Ushwald
- 73
4
votes
1 answer
How to build the smallest regular n-sided polygon that covers an (n-1)-sided polygon?
I want to build a figure that contains seven regular polygons, from a triangle up to a nonagon, where each n-sided polygon covers, with the minimal area possible, the n-1 sided one. An added restriction is that each of them should have a horizontal…
Miguel Farah
- 49
3
votes
1 answer
CAT(0) inequality and CN inequality
I am currently studying CAT(0)-spaces following the book of Bridson and Haefliger: metric spaces of non positive curvature.
Chapter II, exercise 1.9 is the following: Let $(X,d)$ be a geodesic metric space. Then the following are equivalent:
(i) $X$…
Saimel
- 138
3
votes
1 answer
Every riemannian length structure on $\mathbb{R}^n$ is induced by a continuous function $f:\mathbb{R}^n\to \mathbb{E}^n$, to the euclidean space.
This question is a cross post from MathOverflow. Unfortunately the migration of the question is not possible after two months of posting.
I have been reading about length spaces in the (great) book Metric Geometry by Y. Burago, D. Burago and S.…
Dante Grevino
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3
votes
1 answer
Product of two CAT($\kappa$) spaces is CAT($\kappa$) for $\kappa \ge 0$
I would like to see a "metric proof" that if two metric spaces $X$ and $Y$ are CAT($\kappa$) for some $\kappa \ge 0$, then so is their product. I would be satisfied to see a proof for $X=Y=S^2$.
By "metric proof" I mean one which does not rely on…
2
votes
2 answers
Maximum value of two sides of the length of a triangle in a circle, when two points of a triangle are fixed.
For three points $a,b,c$ on a circle, what is the maximum value of $ab+ac$ when points $b$ and $c$ are fixed.
I believe that the maximum value of $ab+ac$ is when $a$ is in the middle of the bigger arc formed by $b$ and $c$..
user413988
2
votes
1 answer
Normal vectors of hedrons
I am trying to quantize the normal vectors I obtain into several bins. To do that, I will compute the dot product of the normal vector with a predefined set of vectors and take the maximum.
My question is: How do I determine the normal vectors of…
Kong
- 884
1
vote
1 answer
Maximal Hausdorff dimension of the inverse image of a point
Let $f \colon [0,1]^2 \to \mathbb{R}$ be an arbitrary continuous function. I was wondering the following:
Does there exist a point $a \in \mathbb{R}$ such that $f^{-1} \{a\}$ has Hausdorff dimension $1$?
If the answer is yes, how small can one make…
J. J.
- 9,432
1
vote
0 answers
vertex representation and half-space representation of a polytope
i'm not a mathematician. So, my question may be stupid. Hope that it won't make you angry.
Is there any mathematical relationship between the vertex representation and the half-space representation of a given polytope?
Clearly speaking, i have the…
noone
- 125
1
vote
1 answer
Fast calculation of the area of intersection between a sphere and a cylinder
In my current research, I am looking at calculating the local porosity of a porous media in cylindrical coordinate (notably, two co-centric cylinders).
To obtain an accurate approximation, I need to calculate the area of intersection between the…
BlaB
- 121
1
vote
3 answers
Area function is continuous on a set of compact sets in $[0,1]^2$
Consider $X=[0,1]^2\subset \mathbb{R}^2$. If $H_X$ is a set of all compact sets in $X$, then we can define a metric $d$ on $H_X$ i.e. Hausdorff metric $d$ :
For $A,\ B\in H_X$, then $d(A,B)=R$ iff there exists a smallest $R$ s.t. $R
HK Lee
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