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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) $d(x,y)+d(y,z) \geq d(x,z)$.

8 votes
1 answer
606 views

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 $\varepsilon > 0$ …
Dominic van der Zypen's user avatar
5 votes
1 answer
412 views

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 $ …
Dominic van der Zypen's user avatar
3 votes
1 answer
82 views

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 …
Dominic van der Zypen's user avatar
2 votes
1 answer
84 views

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( …
Dominic van der Zypen's user avatar
1 vote
1 answer
82 views

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 > …
Dominic van der Zypen's user avatar
1 vote
1 answer
119 views

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 …
Dominic van der Zypen's user avatar
0 votes
1 answer
203 views

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 …
Dominic van der Zypen's user avatar