Mixing
Conditions on Metric Spaces for Equivalence Relations
$begingroup$
Let $V subset mathbb{R}^n$ be a finite volume, and $Ssubset V$. Let $d$ be the usual Euclidean metric on $mathbb{R}^n$:
$$
d(x,y) = sqrt{sum_i (x_i-y_i)^2}
$$
Let $d_S(x)$ denote the distance from $x$ to the (nearest point in) the set $S$:
$$
d_S(x) = min_{yin S} d(x,y)
$$
Now we define a new metric on $mathbb{R}^n$ by
$
D_S(x,y) = d_S(x) + d_S(y)
$
Define the relation $xRy$ by
$$
(x,y) in R Leftrightarrow d(x,y) leq D_S(x,y)
$$
The question: What are the conditions on $S$ (and/or $V$) for
this relation to be an equivalence relation? (It is clearly
reflexive and symmetric: But what conditions for it to be
transitive?)
metric-spaces equivalence-relations
asked Jan 22 at 10:20
Michael Mc GettrickMichael Mc Gettrick
664
$endgroup$
add a comment |
$begingroup$
Let $V subset mathbb{R}^n$ be a finite volume, and $Ssubset V$. Let $d$ be the usual Euclidean metric on $mathbb{R}^n$:
$$
d(x,y) = sqrt{sum_i (x_i-y_i)^2}
$$
Let $d_S(x)$ denote the distance from $x$ to the (nearest point in) the set $S$:
$$
d_S(x) = min_{yin S} d(x,y)
$$
Now we define a new metric on $mathbb{R}^n$ by
$
D_S(x,y) = d_S(x) + d_S(y)
$
Define the relation $xRy$ by
$$
(x,y) in R Leftrightarrow d(x,y) leq D_S(x,y)
$$
The question: What are the conditions on $S$ (and/or $V$) for
this relation to be an equivalence relation? (It is clearly
reflexive and symmetric: But what conditions for it to be
transitive?)
metric-spaces equivalence-relations
asked Jan 22 at 10:20
Michael Mc GettrickMichael Mc Gettrick
664
$endgroup$
$begingroup$
Just realised in above definition, $D_S$ is not a metric (e.g. $D_S(x,x)$ is not always zero). Nonetheless, it is a function on $mathbb{R}^ntimesmathbb{R}^n$, and the question remains about $R$.
$endgroup$
– Michael Mc Gettrick
Jan 24 at 12:04
add a comment |
$begingroup$
Let $V subset mathbb{R}^n$ be a finite volume, and $Ssubset V$. Let $d$ be the usual Euclidean metric on $mathbb{R}^n$:
$$
d(x,y) = sqrt{sum_i (x_i-y_i)^2}
$$
Let $d_S(x)$ denote the distance from $x$ to the (nearest point in) the set $S$:
$$
d_S(x) = min_{yin S} d(x,y)
$$
Now we define a new metric on $mathbb{R}^n$ by
$
D_S(x,y) = d_S(x) + d_S(y)
$
Define the relation $xRy$ by
$$
(x,y) in R Leftrightarrow d(x,y) leq D_S(x,y)
$$
The question: What are the conditions on $S$ (and/or $V$) for
this relation to be an equivalence relation? (It is clearly
reflexive and symmetric: But what conditions for it to be
transitive?)
metric-spaces equivalence-relations
asked Jan 22 at 10:20
Michael Mc GettrickMichael Mc Gettrick
664
$endgroup$
Let $V subset mathbb{R}^n$ be a finite volume, and $Ssubset V$. Let $d$ be the usual Euclidean metric on $mathbb{R}^n$:
$$
d(x,y) = sqrt{sum_i (x_i-y_i)^2}
$$
Let $d_S(x)$ denote the distance from $x$ to the (nearest point in) the set $S$:
$$
d_S(x) = min_{yin S} d(x,y)
$$
Now we define a new metric on $mathbb{R}^n$ by
$
D_S(x,y) = d_S(x) + d_S(y)
$
Define the relation $xRy$ by
$$
(x,y) in R Leftrightarrow d(x,y) leq D_S(x,y)
$$
The question: What are the conditions on $S$ (and/or $V$) for
this relation to be an equivalence relation? (It is clearly
reflexive and symmetric: But what conditions for it to be
transitive?)
metric-spaces equivalence-relations
metric-spaces equivalence-relations
asked Jan 22 at 10:20
Michael Mc GettrickMichael Mc Gettrick
664
asked Jan 22 at 10:20
Michael Mc GettrickMichael Mc Gettrick
664
asked Jan 22 at 10:20
Michael Mc GettrickMichael Mc Gettrick
664
asked Jan 22 at 10:20
Michael Mc GettrickMichael Mc Gettrick
664
asked Jan 22 at 10:20
Michael Mc GettrickMichael Mc Gettrick
664
664
$begingroup$
Just realised in above definition, $D_S$ is not a metric (e.g. $D_S(x,x)$ is not always zero). Nonetheless, it is a function on $mathbb{R}^ntimesmathbb{R}^n$, and the question remains about $R$.
$endgroup$
– Michael Mc Gettrick
Jan 24 at 12:04
add a comment |
$begingroup$
Just realised in above definition, $D_S$ is not a metric (e.g. $D_S(x,x)$ is not always zero). Nonetheless, it is a function on $mathbb{R}^ntimesmathbb{R}^n$, and the question remains about $R$.
$endgroup$
– Michael Mc Gettrick
Jan 24 at 12:04
$begingroup$
Just realised in above definition, $D_S$ is not a metric (e.g. $D_S(x,x)$ is not always zero). Nonetheless, it is a function on $mathbb{R}^ntimesmathbb{R}^n$, and the question remains about $R$.
$endgroup$
– Michael Mc Gettrick
Jan 24 at 12:04
$begingroup$
Just realised in above definition, $D_S$ is not a metric (e.g. $D_S(x,x)$ is not always zero). Nonetheless, it is a function on $mathbb{R}^ntimesmathbb{R}^n$, and the question remains about $R$.
$endgroup$
– Michael Mc Gettrick
Jan 24 at 12:04
add a comment |
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$begingroup$
Just realised in above definition, $D_S$ is not a metric (e.g. $D_S(x,x)$ is not always zero). Nonetheless, it is a function on $mathbb{R}^ntimesmathbb{R}^n$, and the question remains about $R$.
$endgroup$
– Michael Mc Gettrick
Jan 24 at 12:04