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Results tagged with metric-spaces
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user 4312
A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.
10
votes
Non-separable Banach space
Let $\varphi$ be a continuous function supported on $[0,1]$. Then continuum many combinations $\sum c_k \varphi(x+k)$, $c_k\in \{0,1\}$ are separated in our space.
- 109k
10
votes
Accepted
Are two metric spaces isometric if they have the same $\varepsilon$-covering and $\varepsilo...
Certainly no. Consider metric spaces on $n$ points and all distance 1 and 2. There are $2^{n^2/2+o(n^2)}$ such spaces. But only polynomially many different covering and packing functions.
- 109k
6
votes
Accepted
Hausdorff distance and Cauchy sequences
Yes. Choosing a subsequence $n_1<n_2<\dots$ such that $d_H(A_{n_i},A_{n_i+1})\leqslant 2^{-i}$. Then inductively choose points $x_{n_i}\in A_{n_i}$ so that $d(x_{n_i},x_{n_{i+1}})\leqslant 2^{-i}$. It …
- 109k
5
votes
Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$
Denote by $\alpha$, $\beta$, $\gamma$ the angles between $v$ and $x$, $u$ and $x$, $v$ and $u$ respectively (so, $\alpha,\beta,\gamma\in [0,\pi]$). Then $\alpha,\beta,\gamma$ are three planar angles o …
- 109k
3
votes
Accepted
Is every finite graph isomorphic to the proximity graph of some $S\subseteq \mathbb{R}^n$?
Yes. If $n=|V|$, then for small $\varepsilon$ any metric spaces on $n$ points with distances belonging to $\{1-\varepsilon, 1+\varepsilon\} $ is embeddable to $\mathbb{R}^{n-1} $.
- 109k
2
votes
Accepted
Greedy simplices in an ultrametric space (generalized Bhargava $p$-orderings)
It looks that both claims are true.
For $V\subset U$ denote by $f(V)$ the sum of mutual distances between elements of $V$.
Exchange lemma. Let $V\subset U$ and $v\in U$, and let $w$ be a projection …
- 109k
2
votes
Accepted
Right-continuity of covering number
Without additional assumptions on the metric space, it may appear that for every $\varepsilon>1$ the covering number equals 1, but for $\varepsilon=1$ it is infinite. For example, let positive intege …
- 109k
2
votes
Accepted
$d(x,y) = \min\{|x_1−y_1|+|x_2−y_2|, 1−|x_1−y_1|+|x_2−(1−y_2)|\}$ defines a metric on $[0,1)...
It is a metric. Denote $a(x,y):=|x_1−y_1|+|x_2−y_2|$ (a usual $\ell^1$-type metric on the plane), $b(x,y):=1-|x_1-y_1|+|x_2+y_2-1|$. To prove that $d:=\min(a,b)$ is a metric, it suffices to check that …
- 109k