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Results tagged with metric-spaces
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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) $d(x,y)+d(y,z) \geq d(x,z)$.
5
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1
answer
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Spreading $n$ points in $\{0,1\}^n$ as far as possible
Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
We say that a positive integer $s$ is $ …
- 48.9k
3
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1
answer
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Stretching map of $n$ points from $\{0,1\}^n$ to $\{0,1\}^{n+1}$ with respect to their Hammi...
Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
Given an integer $n>0$ and a set $S\su …
- 48.9k
1
vote
1
answer
119
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Is every finite graph isomorphic to the proximity graph of some $S\subseteq \mathbb{R}^n$?
This is the question that I should have asked before asking this older question.
If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ giv …
- 48.9k
2
votes
1
answer
84
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Spatial dimension of a finite graph
If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ given by $V(G(X,d)) = X$ and $$E(G(X,d)) = \big\{\{x,y\}:x\neq y\in X \text{ and } d( …
- 48.9k
0
votes
1
answer
203
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Graphs represented by a subset of a metric space
Let $(X,d)$ be a metric space, and suppose $S\subseteq X$ is a finite subset in which all pairwise distances are distinct (formal definition here).
If $x\in S$ and $k$ is a non-negative integer with …
- 48.9k
1
vote
1
answer
82
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Cauchy subsequences in "Hausdorff Cauchy sets"
This is a follow-up to an older question.
Let $(X,d)$ be a metric space and let $(A_n)_{n\in\mathbb{N}}\subseteq X$ be a sequence of bounded non-empty subsets of $X$ such that for all $\varepsilon > …
- 48.9k
8
votes
1
answer
606
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Hausdorff distance and Cauchy sequences
This is a generalization of an older question.
Let $(X,d)$ be a metric space and let $(A_n)_{n\in\mathbb{N}}\subseteq X$ be a sequence of non-empty closed subsets such that for all $\varepsilon > 0$ …
- 48.9k