All Questions
Tagged with metric-spaces geometry
343
questions
0
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46
views
What, if anything, is this metric on $\mathbb{R}^2$ named? And, what do the open balls in this metric space geometrically look like?
For each $\mathbf{x} := \left( \xi_1, \xi_2 \right) \in \mathbf{R}^2$, let
$$
\lVert \mathbf{x} \rVert := \sqrt{ \xi_1^2 + \xi_2^2 }.
$$
And, for any pair of points $\mathbf{x} := \left( \xi_1, \xi_2 \...
- 26.9k
6
votes
1
answer
89
views
Can the twice-punctured plane be given a homogeneous metric?
A homogeneous metric on a space $X$ is one for which the isometry group acts transitively on its points (for all $x,y\in X$, there is an isometry $\phi$ of $X$ such that $\phi(x)=y$).
If we repeatedly ...
- 3,054
4
votes
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answers
179
views
Shapes with simple distance functions.
Given a set $A$ in $\mathbb{R}^2$, the distance function (DF) of $A$ is defined as
$$
\delta_A(\mathbf{x}) = \inf\{\|\mathbf{x}-\mathbf{y}\|: \mathbf{y} \in A \}
$$
Some sets $A$ have a nice tidy ...
- 43.7k
2
votes
1
answer
140
views
Neighbourhood basis made of balls
I'm studying general topology and a question has come to my mind.
By definition, in a topological space, a neighbourhood basis of a point x is a subset of its neighbourhoods such that every ...
- 579
1
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0
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59
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Metric on $\mathbb{R}^2$ Without Guaranteed Equilateral Triangles
Is there a metric on $\mathbb{R}^2$ (i.e. a function $\mu:\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}^{\geq 0}$ such that $\mu(x,y)= \mu(y,x)$, $\mu(x,y)=0\implies x=y$, and $\mu(x,y)+\mu(y,z)\geq \mu(...
- 1,864
2
votes
1
answer
59
views
What measures allow to distinguish points by their known distances?
Let’s consider five non-colinear points $A$, $B$, $C$, $D$, and $E$ in a normed vector space $\mathbb{R}^4$.
Can we somehow semi-automatically determine if the metric has a following feature—that ...
- 143
0
votes
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answers
57
views
Definition of geodesic in metric spaces
My question is closely related to this:
On the definition of a geodesic in a metric space
I don't understand why in the definition of the geodesic there is the requirement of constant speed. As far as ...
- 2,471
0
votes
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answers
70
views
To prove that the diameter of a solid triangle is the length of the largest side
This is a preliminary in the proof of Cauchy-Goursat Theorem for Triangles in complex analysis,
The solid triangle enclosed by $a,b,c$ in the complex plane is defined as $$\Delta(a,b,c):=\{t_1a+t_2b+...
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16
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Literature search for Isometric boundaries between signed distance functions
I have a bunch of signed distance functions representing non-overlapping objects in 3d space. I would like to find boundaries between the these signed distance functions such that the boundary marks ...
1
vote
0
answers
27
views
Quasi-symmetric maps on the real line and the circle
In some cases, people work with quasi-symmetric maps of $\mathbb{R}$ instead of $\mathbb{S}^1$. More precisely, we say that an increasing homeomorphism $\phi \colon \mathbb{R} \to \mathbb{R}$ is quasi-...
- 125
0
votes
0
answers
25
views
Different definitions of quasi-symmetry and their extensions
One of the possible definitions for quasi-symmetry of a continuous injective map $\phi \colon \mathbb{S}^1 \to \mathbb{C}$ is the following: there exists $M > 0$ such that
$$
1/M \leq \frac{|H(x + ...
- 125
3
votes
2
answers
120
views
Is it possible to define a space and/or a distance function such that there is always more than 1 shortest path between any 2 points?
I am in my second semester of university in maths and physics and thought of a question I am unable to answer.
I asked my analysis teacher of the last semester if it was possible to define a space and/...
- 39
0
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0
answers
47
views
Diameter of small metric balls in a Finsler manifold
Suppose $M$ is a closed manifold with a Finsler metric $F$, and let $d$ be the induced distance on $M$. In general, due to the asymmetry of $F$, the distance $d$ is asymmetric as well.
Consider the ...
- 2,226
2
votes
1
answer
53
views
Edges of a $K_4$ cannot be too short
This problem arised while I was working through Erdös-Bollobas's solution for Ramsey-Turán Problem however i believe that details for that isn't necessary.
Consider the following construction
\begin{...
1
vote
0
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58
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Calculating Cosine Similarity Between Two Points in Hyperspherical Coordinate
I'm working with points in an n-dimensional hyperspherical coordinate system, in other words, my points are in the shape $(r, \theta_1, \theta_2, ..., \theta_{n-1})$. I want to calculate the angle ...
- 770