Tagged Questions

0
votes
1answer
38 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 ...
1
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0answers
70 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 ...
7
votes
1answer
336 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
144 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 ...
0
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0answers
63 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) $X$ ...
2
votes
2answers
76 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 $f^{-1}$. ...
9
votes
3answers
351 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 ...
1
vote
1answer
820 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 ...
4
votes
5answers
484 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
256 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
113 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
60 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 ...
4
votes
1answer
185 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
454 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
143 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
204 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 ...