Integration over convex polytope
$begingroup$
I have a small problem (as stated below). Moreover, I am new to this site (asking questions that is) and not a mathematician by trade. I tried to be as precise as possible in asking the question. If anything is unclear, please let me know and I will try to make my point more clear.
The problem is the following: I have a convex polytope $A$ which is split by a half space into the two convex polytopes $A_L$ and $A_R$. Moreover, I have a continuous function $p(x)$ and a probability density function $f(X)$, where $X$ is an $i.i.d.$ random vector (with mutually independent elements) and I know that
$$
frac{int_Ap(x)f(x)dx}{int_Af(x)dx}=frac{int_{A_L}p(x)f(x)dx}{int_{A_L}f(x)dx}=frac{int_{A_R}p(x)f(x)dx}{int_{A_R}f(x)dx}
$$
holds for all possible splits of $A$ into $A_L$ and $A_R$.
I want to show that if the above equality holds for all possible splits of $A$ into $A_L$ and $A_R$, then $p(x)$ must be constant on $A$.
I can show the result in the case of a scalar r.v. $X$. However, I struggle to show the result for the general case where $X$ is an $i.i.d.$ random vector (with mutually independent elements). However, I also realize that the problem itself is independent of the fact that $f(x)$ is a pdf. Therefore, I was thinking that there might be a general mathematical result which can be used to show the claim.
I very much appreciate any pointers towards a way to prove the above statement.
real-analysis probability integration
asked Dec 13 '18 at 9:29
rothemrothem
262
$endgroup$
add a comment |
$begingroup$
I have a small problem (as stated below). Moreover, I am new to this site (asking questions that is) and not a mathematician by trade. I tried to be as precise as possible in asking the question. If anything is unclear, please let me know and I will try to make my point more clear.
The problem is the following: I have a convex polytope $A$ which is split by a half space into the two convex polytopes $A_L$ and $A_R$. Moreover, I have a continuous function $p(x)$ and a probability density function $f(X)$, where $X$ is an $i.i.d.$ random vector (with mutually independent elements) and I know that
$$
frac{int_Ap(x)f(x)dx}{int_Af(x)dx}=frac{int_{A_L}p(x)f(x)dx}{int_{A_L}f(x)dx}=frac{int_{A_R}p(x)f(x)dx}{int_{A_R}f(x)dx}
$$
holds for all possible splits of $A$ into $A_L$ and $A_R$.
I want to show that if the above equality holds for all possible splits of $A$ into $A_L$ and $A_R$, then $p(x)$ must be constant on $A$.
I can show the result in the case of a scalar r.v. $X$. However, I struggle to show the result for the general case where $X$ is an $i.i.d.$ random vector (with mutually independent elements). However, I also realize that the problem itself is independent of the fact that $f(x)$ is a pdf. Therefore, I was thinking that there might be a general mathematical result which can be used to show the claim.
I very much appreciate any pointers towards a way to prove the above statement.
real-analysis probability integration
asked Dec 13 '18 at 9:29
rothemrothem
262
$endgroup$
add a comment |
$begingroup$
I have a small problem (as stated below). Moreover, I am new to this site (asking questions that is) and not a mathematician by trade. I tried to be as precise as possible in asking the question. If anything is unclear, please let me know and I will try to make my point more clear.
The problem is the following: I have a convex polytope $A$ which is split by a half space into the two convex polytopes $A_L$ and $A_R$. Moreover, I have a continuous function $p(x)$ and a probability density function $f(X)$, where $X$ is an $i.i.d.$ random vector (with mutually independent elements) and I know that
$$
frac{int_Ap(x)f(x)dx}{int_Af(x)dx}=frac{int_{A_L}p(x)f(x)dx}{int_{A_L}f(x)dx}=frac{int_{A_R}p(x)f(x)dx}{int_{A_R}f(x)dx}
$$
holds for all possible splits of $A$ into $A_L$ and $A_R$.
I want to show that if the above equality holds for all possible splits of $A$ into $A_L$ and $A_R$, then $p(x)$ must be constant on $A$.
I can show the result in the case of a scalar r.v. $X$. However, I struggle to show the result for the general case where $X$ is an $i.i.d.$ random vector (with mutually independent elements). However, I also realize that the problem itself is independent of the fact that $f(x)$ is a pdf. Therefore, I was thinking that there might be a general mathematical result which can be used to show the claim.
I very much appreciate any pointers towards a way to prove the above statement.
real-analysis probability integration
asked Dec 13 '18 at 9:29
rothemrothem
262
$endgroup$
I have a small problem (as stated below). Moreover, I am new to this site (asking questions that is) and not a mathematician by trade. I tried to be as precise as possible in asking the question. If anything is unclear, please let me know and I will try to make my point more clear.
The problem is the following: I have a convex polytope $A$ which is split by a half space into the two convex polytopes $A_L$ and $A_R$. Moreover, I have a continuous function $p(x)$ and a probability density function $f(X)$, where $X$ is an $i.i.d.$ random vector (with mutually independent elements) and I know that
$$
frac{int_Ap(x)f(x)dx}{int_Af(x)dx}=frac{int_{A_L}p(x)f(x)dx}{int_{A_L}f(x)dx}=frac{int_{A_R}p(x)f(x)dx}{int_{A_R}f(x)dx}
$$
holds for all possible splits of $A$ into $A_L$ and $A_R$.
I want to show that if the above equality holds for all possible splits of $A$ into $A_L$ and $A_R$, then $p(x)$ must be constant on $A$.
I can show the result in the case of a scalar r.v. $X$. However, I struggle to show the result for the general case where $X$ is an $i.i.d.$ random vector (with mutually independent elements). However, I also realize that the problem itself is independent of the fact that $f(x)$ is a pdf. Therefore, I was thinking that there might be a general mathematical result which can be used to show the claim.
I very much appreciate any pointers towards a way to prove the above statement.
real-analysis probability integration
real-analysis probability integration
asked Dec 13 '18 at 9:29
rothemrothem
262
asked Dec 13 '18 at 9:29
rothemrothem
262
asked Dec 13 '18 at 9:29
rothemrothem
262
asked Dec 13 '18 at 9:29
rothemrothem
262
asked Dec 13 '18 at 9:29
rothemrothem
262
262
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