Increasing multivariate function: interpreting the definition

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I have doubts on how to interpret the following definition of $L$-increasing function.

Let $barmathbbRequiv mathbbRcup -infty, +infty$, where $mathbbR$ denotes the real line.

Let $barmathbbR^L$ be the $L$-fold Cartesian product for a positive integer $L$.

Let $mathcalUequiv mathcalU_1 times ... times mathcalU_L subseteq barmathbbR^L$ with $ mathcalU_lsubseteq barmathbbR$ $forall l in 1,...,L$.

Definition $F: mathcalUrightarrow ...$ is $L$-increasing if $forall u', u''in mathcalU$ with $u'leq u''$ component-wise

$$
textVol_F(u', u'')equiv sum_uin textVrt(u', u'')textsgn_u', u''(u)F(u)geq 0
$$
where

• $textVrt(u', u'')equiv uin mathcalU text: u_lin u_l', u_l'' text forall lin 1,...,L$




• $textsgn_u', u''(u)equiv begincases
  1 & text if $u_l=u_l'$ for an even number of $lin 1,...,L$\
  -1 & text if $u_l=u_l'$ for an odd number of $lin 1,...,L$\
  endcases$




• $textVol_F(u', u'')$ is the $F$-volume of the $L$-box $[u_1', u_1'']times ... times [u_L', u_L'']$ and the elements of the set $textVrt(u', u'')$ are the vertices of the $L$-box

Question I am confused on the last part of the definition: "$textVol_F(u', u'')$ is the $F$-volume of the $L$-box $[u_1', u_1'']times ... times [u_L', u_L'']$ and the elements of the set $textVrt(u', u'')$ are the vertices of the $L$-box". Could you help me to understand why $textVol_F(u', u'')$ is a volume?

I have tried to picture an example for $L=2$

The green area is $F$.

Suppose $u'equiv (2,2)$ and $u''equiv (4,3)$. Hence, $textVrt(u', u'')equiv u', u'', A, B$ with $Aequiv (u'_1, u''_2)$ and $Bequiv (u''_1, u'_2)$. Thus, we get

$$
textVol_F(u', u'')equiv 1*F(u')+1*F(u'')-1*F(A)-1*F(B)
$$

Why this is a volume? It looks like just the difference between points' heights.

geometry functional-analysis functions

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asked Jul 24 at 10:14

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I have doubts on how to interpret the following definition of $L$-increasing function.

Let $barmathbbRequiv mathbbRcup -infty, +infty$, where $mathbbR$ denotes the real line.

Let $barmathbbR^L$ be the $L$-fold Cartesian product for a positive integer $L$.

Let $mathcalUequiv mathcalU_1 times ... times mathcalU_L subseteq barmathbbR^L$ with $ mathcalU_lsubseteq barmathbbR$ $forall l in 1,...,L$.

Definition $F: mathcalUrightarrow ...$ is $L$-increasing if $forall u', u''in mathcalU$ with $u'leq u''$ component-wise

$$
textVol_F(u', u'')equiv sum_uin textVrt(u', u'')textsgn_u', u''(u)F(u)geq 0
$$
where

• $textVrt(u', u'')equiv uin mathcalU text: u_lin u_l', u_l'' text forall lin 1,...,L$




• $textsgn_u', u''(u)equiv begincases
  1 & text if $u_l=u_l'$ for an even number of $lin 1,...,L$\
  -1 & text if $u_l=u_l'$ for an odd number of $lin 1,...,L$\
  endcases$




• $textVol_F(u', u'')$ is the $F$-volume of the $L$-box $[u_1', u_1'']times ... times [u_L', u_L'']$ and the elements of the set $textVrt(u', u'')$ are the vertices of the $L$-box

Question I am confused on the last part of the definition: "$textVol_F(u', u'')$ is the $F$-volume of the $L$-box $[u_1', u_1'']times ... times [u_L', u_L'']$ and the elements of the set $textVrt(u', u'')$ are the vertices of the $L$-box". Could you help me to understand why $textVol_F(u', u'')$ is a volume?

I have tried to picture an example for $L=2$

The green area is $F$.

Suppose $u'equiv (2,2)$ and $u''equiv (4,3)$. Hence, $textVrt(u', u'')equiv u', u'', A, B$ with $Aequiv (u'_1, u''_2)$ and $Bequiv (u''_1, u'_2)$. Thus, we get

$$
textVol_F(u', u'')equiv 1*F(u')+1*F(u'')-1*F(A)-1*F(B)
$$

Why this is a volume? It looks like just the difference between points' heights.

geometry functional-analysis functions

share|cite|improve this question

asked Jul 24 at 10:14

TEX

2419

add a comment | 

up vote
4
down vote

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up vote
4
down vote

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1

I have doubts on how to interpret the following definition of $L$-increasing function.

Let $barmathbbRequiv mathbbRcup -infty, +infty$, where $mathbbR$ denotes the real line.

Let $barmathbbR^L$ be the $L$-fold Cartesian product for a positive integer $L$.

Let $mathcalUequiv mathcalU_1 times ... times mathcalU_L subseteq barmathbbR^L$ with $ mathcalU_lsubseteq barmathbbR$ $forall l in 1,...,L$.

Definition $F: mathcalUrightarrow ...$ is $L$-increasing if $forall u', u''in mathcalU$ with $u'leq u''$ component-wise

$$
textVol_F(u', u'')equiv sum_uin textVrt(u', u'')textsgn_u', u''(u)F(u)geq 0
$$
where

• $textVrt(u', u'')equiv uin mathcalU text: u_lin u_l', u_l'' text forall lin 1,...,L$




• $textsgn_u', u''(u)equiv begincases
  1 & text if $u_l=u_l'$ for an even number of $lin 1,...,L$\
  -1 & text if $u_l=u_l'$ for an odd number of $lin 1,...,L$\
  endcases$




• $textVol_F(u', u'')$ is the $F$-volume of the $L$-box $[u_1', u_1'']times ... times [u_L', u_L'']$ and the elements of the set $textVrt(u', u'')$ are the vertices of the $L$-box

Question I am confused on the last part of the definition: "$textVol_F(u', u'')$ is the $F$-volume of the $L$-box $[u_1', u_1'']times ... times [u_L', u_L'']$ and the elements of the set $textVrt(u', u'')$ are the vertices of the $L$-box". Could you help me to understand why $textVol_F(u', u'')$ is a volume?

I have tried to picture an example for $L=2$

The green area is $F$.

Suppose $u'equiv (2,2)$ and $u''equiv (4,3)$. Hence, $textVrt(u', u'')equiv u', u'', A, B$ with $Aequiv (u'_1, u''_2)$ and $Bequiv (u''_1, u'_2)$. Thus, we get

$$
textVol_F(u', u'')equiv 1*F(u')+1*F(u'')-1*F(A)-1*F(B)
$$

Why this is a volume? It looks like just the difference between points' heights.

geometry functional-analysis functions

share|cite|improve this question

asked Jul 24 at 10:14

TEX

2419

I have doubts on how to interpret the following definition of $L$-increasing function.

Let $barmathbbRequiv mathbbRcup -infty, +infty$, where $mathbbR$ denotes the real line.

Let $barmathbbR^L$ be the $L$-fold Cartesian product for a positive integer $L$.

Let $mathcalUequiv mathcalU_1 times ... times mathcalU_L subseteq barmathbbR^L$ with $ mathcalU_lsubseteq barmathbbR$ $forall l in 1,...,L$.

Definition $F: mathcalUrightarrow ...$ is $L$-increasing if $forall u', u''in mathcalU$ with $u'leq u''$ component-wise

$$
textVol_F(u', u'')equiv sum_uin textVrt(u', u'')textsgn_u', u''(u)F(u)geq 0
$$
where

• $textVrt(u', u'')equiv uin mathcalU text: u_lin u_l', u_l'' text forall lin 1,...,L$




• $textsgn_u', u''(u)equiv begincases
  1 & text if $u_l=u_l'$ for an even number of $lin 1,...,L$\
  -1 & text if $u_l=u_l'$ for an odd number of $lin 1,...,L$\
  endcases$




• $textVol_F(u', u'')$ is the $F$-volume of the $L$-box $[u_1', u_1'']times ... times [u_L', u_L'']$ and the elements of the set $textVrt(u', u'')$ are the vertices of the $L$-box

Question I am confused on the last part of the definition: "$textVol_F(u', u'')$ is the $F$-volume of the $L$-box $[u_1', u_1'']times ... times [u_L', u_L'']$ and the elements of the set $textVrt(u', u'')$ are the vertices of the $L$-box". Could you help me to understand why $textVol_F(u', u'')$ is a volume?

I have tried to picture an example for $L=2$

The green area is $F$.

Suppose $u'equiv (2,2)$ and $u''equiv (4,3)$. Hence, $textVrt(u', u'')equiv u', u'', A, B$ with $Aequiv (u'_1, u''_2)$ and $Bequiv (u''_1, u'_2)$. Thus, we get

$$
textVol_F(u', u'')equiv 1*F(u')+1*F(u'')-1*F(A)-1*F(B)
$$

Why this is a volume? It looks like just the difference between points' heights.

geometry functional-analysis functions

share|cite|improve this question

asked Jul 24 at 10:14

TEX

2419

share|cite|improve this question

share|cite|improve this question

asked Jul 24 at 10:14

TEX

2419

asked Jul 24 at 10:14

TEX

2419

asked Jul 24 at 10:14

TEX

2419

2419

add a comment | 

add a comment | 

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