Metrics on families of functions
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Let $mathcal{F}$ be a family of functions $Dsubseteq mathbb{R}^nrightarrow mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally associate with $mathcal{F}$, namely $C^p$ metrics, $L^p$ metrics and metrics that involve Lipschitz constants.
I'd like to hear about other metrics --possibly less well known than the ones mentioned-- that we associate naturally* with families of functions with certain properties.
*By naturally I mean that the definition of the metric has something to do with the functions themselves, so not just metrics that you may consider say on any set of a given cardinality.
general-topology functional-analysis metric-spaces big-list
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add a comment |
$begingroup$
Let $mathcal{F}$ be a family of functions $Dsubseteq mathbb{R}^nrightarrow mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally associate with $mathcal{F}$, namely $C^p$ metrics, $L^p$ metrics and metrics that involve Lipschitz constants.
I'd like to hear about other metrics --possibly less well known than the ones mentioned-- that we associate naturally* with families of functions with certain properties.
*By naturally I mean that the definition of the metric has something to do with the functions themselves, so not just metrics that you may consider say on any set of a given cardinality.
general-topology functional-analysis metric-spaces big-list
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1
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The Sobolev metrics certainly form an important class of examples.
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– pseudocydonia
Oct 29 '18 at 3:17
add a comment |
$begingroup$
Let $mathcal{F}$ be a family of functions $Dsubseteq mathbb{R}^nrightarrow mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally associate with $mathcal{F}$, namely $C^p$ metrics, $L^p$ metrics and metrics that involve Lipschitz constants.
I'd like to hear about other metrics --possibly less well known than the ones mentioned-- that we associate naturally* with families of functions with certain properties.
*By naturally I mean that the definition of the metric has something to do with the functions themselves, so not just metrics that you may consider say on any set of a given cardinality.
general-topology functional-analysis metric-spaces big-list
$endgroup$
Let $mathcal{F}$ be a family of functions $Dsubseteq mathbb{R}^nrightarrow mathbb{R}$. Depending on the characteristics of these functions there are a number of metrics that we would naturally associate with $mathcal{F}$, namely $C^p$ metrics, $L^p$ metrics and metrics that involve Lipschitz constants.
I'd like to hear about other metrics --possibly less well known than the ones mentioned-- that we associate naturally* with families of functions with certain properties.
*By naturally I mean that the definition of the metric has something to do with the functions themselves, so not just metrics that you may consider say on any set of a given cardinality.
general-topology functional-analysis metric-spaces big-list
general-topology functional-analysis metric-spaces big-list
edited Dec 6 '18 at 15:18
Anguepa
asked Oct 16 '18 at 15:16
AnguepaAnguepa
1,406819
1,406819
1
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The Sobolev metrics certainly form an important class of examples.
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– pseudocydonia
Oct 29 '18 at 3:17
add a comment |
1
$begingroup$
The Sobolev metrics certainly form an important class of examples.
$endgroup$
– pseudocydonia
Oct 29 '18 at 3:17
1
1
$begingroup$
The Sobolev metrics certainly form an important class of examples.
$endgroup$
– pseudocydonia
Oct 29 '18 at 3:17
$begingroup$
The Sobolev metrics certainly form an important class of examples.
$endgroup$
– pseudocydonia
Oct 29 '18 at 3:17
add a comment |
1 Answer
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$begingroup$
There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as
$$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$
Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.
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$begingroup$
There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as
$$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$
Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.
$endgroup$
add a comment |
$begingroup$
There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as
$$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$
Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.
$endgroup$
add a comment |
$begingroup$
There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as
$$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$
Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.
$endgroup$
There are a number of interesting metrics on (metric) measure spaces, the simplest of which might be the discrepancy metric. Let $(Omega, rho)$ be a metric space and let $mu$ and $nu$ be two probability measures defined on $Omega$. Let ${cal B}$ be the set of all closed balls in $Omega$. Then the discrepancy metric is defined as
$$d_D(mu, nu) := sup_{B in {cal B}} |mu(B) -nu(B)|$$
Since $mu$ and $nu$ are probability measures we see immediately that $0leq d_D leq 1$, and (more importantly) that it is scale-invariant -- multiplying $rho$ by any positive constant does not affect $d_D$. It's commonly used to study random walks on groups.
answered Dec 6 '18 at 20:48
postmortespostmortes
1,89321117
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The Sobolev metrics certainly form an important class of examples.
$endgroup$
– pseudocydonia
Oct 29 '18 at 3:17