Average distance between two points
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Let $Ssubsetmathbb{R}^n$ be a bounded, convex set. Consider the problem of finding the average distance between two randomly chosen points from $S$: $$d = mathbb{E}_{asim S}[mathbb{E}_{bsim S}[d(a, b)]].$$ Is it true that this average is achieved for every fixed point: that is, if we fix $a$, do we get the same result? Does $mathbb{E}_{bin S}[d(a, b)] = d$ for every fixed $ain S$?
probability probability-theory
asked Dec 16 '18 at 22:39
user114743user114743
1046
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add a comment |
$begingroup$
Let $Ssubsetmathbb{R}^n$ be a bounded, convex set. Consider the problem of finding the average distance between two randomly chosen points from $S$: $$d = mathbb{E}_{asim S}[mathbb{E}_{bsim S}[d(a, b)]].$$ Is it true that this average is achieved for every fixed point: that is, if we fix $a$, do we get the same result? Does $mathbb{E}_{bin S}[d(a, b)] = d$ for every fixed $ain S$?
probability probability-theory
asked Dec 16 '18 at 22:39
user114743user114743
1046
$endgroup$
1
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I have no ideea about the problem, but it feels weird that it got two upvotes in less than $3$ minutes to both question & answer and no one upvoted afterwards for the past $10$ minutes.
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– Zacky
Dec 16 '18 at 22:51
add a comment |
$begingroup$
Let $Ssubsetmathbb{R}^n$ be a bounded, convex set. Consider the problem of finding the average distance between two randomly chosen points from $S$: $$d = mathbb{E}_{asim S}[mathbb{E}_{bsim S}[d(a, b)]].$$ Is it true that this average is achieved for every fixed point: that is, if we fix $a$, do we get the same result? Does $mathbb{E}_{bin S}[d(a, b)] = d$ for every fixed $ain S$?
probability probability-theory
asked Dec 16 '18 at 22:39
user114743user114743
1046
$endgroup$
Let $Ssubsetmathbb{R}^n$ be a bounded, convex set. Consider the problem of finding the average distance between two randomly chosen points from $S$: $$d = mathbb{E}_{asim S}[mathbb{E}_{bsim S}[d(a, b)]].$$ Is it true that this average is achieved for every fixed point: that is, if we fix $a$, do we get the same result? Does $mathbb{E}_{bin S}[d(a, b)] = d$ for every fixed $ain S$?
probability probability-theory
probability probability-theory
asked Dec 16 '18 at 22:39
user114743user114743
1046
asked Dec 16 '18 at 22:39
user114743user114743
1046
asked Dec 16 '18 at 22:39
user114743user114743
1046
asked Dec 16 '18 at 22:39
user114743user114743
1046
asked Dec 16 '18 at 22:39
user114743user114743
1046
1046
1
$begingroup$
I have no ideea about the problem, but it feels weird that it got two upvotes in less than $3$ minutes to both question & answer and no one upvoted afterwards for the past $10$ minutes.
$endgroup$
– Zacky
Dec 16 '18 at 22:51
add a comment |
1
$begingroup$
I have no ideea about the problem, but it feels weird that it got two upvotes in less than $3$ minutes to both question & answer and no one upvoted afterwards for the past $10$ minutes.
$endgroup$
– Zacky
Dec 16 '18 at 22:51
1
1
$begingroup$
I have no ideea about the problem, but it feels weird that it got two upvotes in less than $3$ minutes to both question & answer and no one upvoted afterwards for the past $10$ minutes.
$endgroup$
– Zacky
Dec 16 '18 at 22:51
$begingroup$
I have no ideea about the problem, but it feels weird that it got two upvotes in less than $3$ minutes to both question & answer and no one upvoted afterwards for the past $10$ minutes.
$endgroup$
– Zacky
Dec 16 '18 at 22:51
add a comment |
1 Answer
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I guess this would be an obvious counterexample, can someone confirm if this is correct? If $S$ is a line of length $L$, then the average distance of two randomly chosen points is $L/3$. But if one of the points was fixed at an endpoint, the average distance would be $L/2$...
answered Dec 16 '18 at 22:40
user114743user114743
1046
$endgroup$
add a comment |
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$begingroup$
I guess this would be an obvious counterexample, can someone confirm if this is correct? If $S$ is a line of length $L$, then the average distance of two randomly chosen points is $L/3$. But if one of the points was fixed at an endpoint, the average distance would be $L/2$...
answered Dec 16 '18 at 22:40
user114743user114743
1046
$endgroup$
add a comment |
$begingroup$
I guess this would be an obvious counterexample, can someone confirm if this is correct? If $S$ is a line of length $L$, then the average distance of two randomly chosen points is $L/3$. But if one of the points was fixed at an endpoint, the average distance would be $L/2$...
answered Dec 16 '18 at 22:40
user114743user114743
1046
$endgroup$
add a comment |
$begingroup$
I guess this would be an obvious counterexample, can someone confirm if this is correct? If $S$ is a line of length $L$, then the average distance of two randomly chosen points is $L/3$. But if one of the points was fixed at an endpoint, the average distance would be $L/2$...
answered Dec 16 '18 at 22:40
user114743user114743
1046
$endgroup$
I guess this would be an obvious counterexample, can someone confirm if this is correct? If $S$ is a line of length $L$, then the average distance of two randomly chosen points is $L/3$. But if one of the points was fixed at an endpoint, the average distance would be $L/2$...
answered Dec 16 '18 at 22:40
user114743user114743
1046
answered Dec 16 '18 at 22:40
user114743user114743
1046
answered Dec 16 '18 at 22:40
user114743user114743
1046
answered Dec 16 '18 at 22:40
user114743user114743
1046
1046
add a comment |
add a comment |
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I have no ideea about the problem, but it feels weird that it got two upvotes in less than $3$ minutes to both question & answer and no one upvoted afterwards for the past $10$ minutes.
$endgroup$
– Zacky
Dec 16 '18 at 22:51