1
vote
2answers
50 views

Looking for a proof that the diameter of the smallest bounding circle is less than or equal to $\frac{2}{\sqrt{3}}$ times the diameter of the set

This came up while I was attempting to solve an old journal problem. It's not the easiest result to search for so I figured I would ask. Let $E$ be a subset of $\mathbb{R}^2$, then the diameter of ...
1
vote
0answers
27 views

In regards to metric spaces, does $d^\star$ have an accepted name, or notation? Do any authors use it?

(I write $\omega$ for the set $\{0,1,2,\ldots\}$.) Let $X$ denote a metric space with metric $d$. Define a function $d^{\star} : X^\omega \times X^\omega \rightarrow [0,\infty]^\omega$ by writing ...
1
vote
1answer
44 views

Gromov hyperbolic metric spaces are quasi-convex

I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a ...
2
votes
0answers
58 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
1
vote
1answer
71 views

What is the metric spaces needed to motivate concepts of general topology?

I intend to start learning some topology on my own. I wonder How much metric spaces I should know in order to motivate the concepts of topology? I know it's possible to learn topology without any ...
3
votes
0answers
23 views

Is there a name for this partial order between metrics?

Suppose we have a set $X$ and two metrics $d_1,d_2$ on it (which may or may not attain $\infty$). Assume furthermore that $d_1,d_2$ have the same metric components (where a metric comoponent is a ...
1
vote
1answer
53 views

Proving that a function is a metric

Let $$p(x,y)= \left|\frac{1}{x} - \frac{1}{y}\right|$$ for $x,y > 0$. Prove that $p$ is a metric for $(0,\infty)$. This question is from Methods of Real Analysis, 2nd edition by Richard ...
2
votes
0answers
63 views

An (extended) semimetric on surfaces

Given a smooth surface $S \subseteq \mathbb{R}^3$, like the surface of sphere, we can define the following extended semimetric $d : S^2 \to [0, \infty]$, where $$ d(x,y) = \inf\{\lVert x - p\rVert + ...
6
votes
2answers
82 views

Has the idea of generalizing the codomain of a metric been seriously considered?

The long line is much longer than $\mathbb{R}$, and indeed many chains have this property. Thus, since metrics are usually assumed to be real-valued, this can be understood as an assumption that ...
1
vote
1answer
102 views

What is the best way to define the diameter of the empty subset of a metric space?

This question is related to Why are metric spaces non-empty? . I think that a metric space should allowed to be empty, and many authorities, including Rudin, agree with me. That way, any subset of a ...
2
votes
1answer
78 views

Different “$\pi$s” [duplicate]

Does any one know of a concept analogous to $\pi$ in metric spaces. Namely, taking the all the points $1$ away from a point, and measuring the distance as some sort of limit? This was prompted when I ...
0
votes
1answer
35 views

Existence of a geodesic in a complete separable metric space

If I have $X$ a complete separable metric space, $x, y \in X$ arbitrary points, how can I define a constant speed geodesic, i.e. a continuous map $g : [0,1] \rightarrow X$ such that $$ d(g(t), g(s)) = ...
2
votes
2answers
118 views

book for metric spaces

Can anybody suggest me a good book on Metric Spaces. Although I am not new to this subject, but want to polish my knowledge. I want a book which can clearly clear my basics. I want to start from the ...
4
votes
2answers
136 views

Distance metric for (infinite) lines in 3D

I would like a metric $d(\;)$ between pairs of (infinite) lines in $\mathbb{R}^3$, with these properties: If two lines $L_1$ and $L_2$ are parallel and separated by distance $x$, then $d(L_1,L_2) = ...
3
votes
1answer
197 views

Extending a function beyond the completion/closure of its domain

In analysis there are certain theorems that tell under which conditions you can continuously extend a continuous functions to the closure/completion of its domain (which actually give the same set, ...
0
votes
1answer
56 views

Metric Spaces needed for Differential Geometry

I've asked here about some texts about differential geometry which doesn't assumes that the reader knows general topology. I've got good references as Do Carmo's Differential Geometry of Curves and ...
6
votes
3answers
196 views

Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space?

I have question. Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space ? Please can you give me an advice some book names? Thank you!
2
votes
0answers
81 views

A question about a metric on $\mathbb{R}^\mathbb{N}$

Consider the metric space $(\mathbb{R}^{\mathbb{N}},d)$ where for $x,y\in\mathbb{R}^\mathbb{N}$ $$ d(x,y) = \sum_{n=1}^{\infty} 2^{- n} \frac{\bigvee_{k\leq n}\left|x_k-y_k\right|}{1 + \bigvee_{k\leq ...
8
votes
1answer
393 views

Algorithms for computing or numerically approximating the Prokhorov metric?

I am interested in the following practical question: Given two measures (say those of two parametric distributions), is there an algorithm for computing the Prokhorov metric between them? The general ...
5
votes
3answers
158 views

How to define a “metric” whose range is not the reals?

This may sound a very stupid question. Why do we need to restrict a metric from a general set $X$ to map to the positive real numbers? I try to be clearer. We are given a set $X$ and a totally ...
2
votes
0answers
80 views

Reference for the characterization of completeness for metric spaces

I have the following criteria for the completeness of a metric space that I want to use in some research paper. Let $(X,d)$ be a metric space. The following conditions are equivalent: (1) $X$ ...
3
votes
2answers
79 views

Entropy of a North South Transformation.

Let $f:\mathbb{S}^2\to\mathbb{S}^2$ be a continuous north south Transformation, in other words, the point $(0,0,1)$ is a global attractor for $f$ and $(0,0,-1)$ is a global attractor for $f^{-1}$. ...
10
votes
3answers
711 views

Compactness of the Grassmannian

Let $V$ be a finite-dimensional inner product space. For $0 \leq d \leq \text{dim}(V)$, define the Grassmannian $G(V, d)$ to be the set of all $d$-dimensional linear subspaces of $V$, equipped with ...
8
votes
2answers
2k views

Cosine similarity / distance and triangle equation

There is a similarity function particular popular for processing sparse vectors such as textual data (word frequency counts etc.) commonly referred to as cosine similarity. There are two variants to ...
5
votes
5answers
1k views

A good book for metric spaces?

I'm looking for some book to study metric spaces. 2 years ago, used a book called Burkill, as well as using multiple topological concepts, I have also studied the Munkres, Chapters 2,3,4,5,6,9. ...
4
votes
1answer
317 views

Completion of Topological Group with Metric

Related to this question, I'm having trouble understanding the construction of the completion of a topological group with metric structure. In particular, under what conditions is the completion also ...
1
vote
1answer
163 views

Voronoi diagram with different metric functions

Given a metric space $(X,d)$ and finite number of points $(x_i)_{i=1}^n$ the Voronoi diagram (or the Dirichlet cell) $C_i$ is given by $$ C_i = \{x\in X:d(x,x_i)<\min\limits_{j\neq i}d(x,x_j)\}. ...
0
votes
1answer
69 views

Low distortion embeddings (reference request)

I read about the Johnson Lindenstrauss Lemma. I googled and found that low distortion embeddings is a live subject, but it seems that many interesting results are already known. Is there a book on ...
5
votes
1answer
227 views

Why has the extreme value function never been defined? –Or has it?

Just as when x and y are arbitrary real numbers, we often wish to consider their distance apart, and use the absolute value function to do so (namely, by means of the expression |x – y|), so also when ...
14
votes
1answer
644 views

A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded

The title is a claim my classmate made during our summer vacation :D He showed me a TeX file describing a proof of his claim, and it contains a fairly short but elegant proof. He says that the ...
4
votes
1answer
147 views

Projection to space with smaller dimension that saves a distance

Suppose we have a countable set of objects $\{x_i|i \in [1..m]\}$ in a metric space $(\mathbb R^n,d_n)$ and a map ($F$) mapping the objects to objects in a metric space $(\mathbb R^1,d_1)$. For each ...
1
vote
2answers
233 views

Gradient flows in metric spaces

What is a good introduction in gradient flows in metric spaces? I know the book Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe ...