Can the constant $3$ in the Vitali covering lemma be replaced by any positive constant less than that in the...
I am stuck with a question that the constant $3$ in the Vitali covering lemma can not be replaced by any positive constant less than that in the finite case.
Observe that this question is different from the question of why $3$ is considered as bad constant because I am not considering infinite case here.
For example, I was thinking in real line and the argument can be generalised in the higher dimension; if I have $(0,4)$ and constant is $c<3$ then taking the interval $(8,4+frac{3-c}{2})$ or anything would not give me my answer. So I was doubting the validity of the statement in a metric space but I don't know whether I am missing anything or not!!
real-analysis general-topology measure-theory metric-spaces real-numbers
edited Nov 10 '18 at 9:16
Christian Blatter
172k7113326
asked Nov 10 '18 at 8:53
GimgimGimgim
1629
add a comment |
I am stuck with a question that the constant $3$ in the Vitali covering lemma can not be replaced by any positive constant less than that in the finite case.
Observe that this question is different from the question of why $3$ is considered as bad constant because I am not considering infinite case here.
For example, I was thinking in real line and the argument can be generalised in the higher dimension; if I have $(0,4)$ and constant is $c<3$ then taking the interval $(8,4+frac{3-c}{2})$ or anything would not give me my answer. So I was doubting the validity of the statement in a metric space but I don't know whether I am missing anything or not!!
real-analysis general-topology measure-theory metric-spaces real-numbers
edited Nov 10 '18 at 9:16
Christian Blatter
172k7113326
asked Nov 10 '18 at 8:53
GimgimGimgim
1629
add a comment |
I am stuck with a question that the constant $3$ in the Vitali covering lemma can not be replaced by any positive constant less than that in the finite case.
Observe that this question is different from the question of why $3$ is considered as bad constant because I am not considering infinite case here.
For example, I was thinking in real line and the argument can be generalised in the higher dimension; if I have $(0,4)$ and constant is $c<3$ then taking the interval $(8,4+frac{3-c}{2})$ or anything would not give me my answer. So I was doubting the validity of the statement in a metric space but I don't know whether I am missing anything or not!!
real-analysis general-topology measure-theory metric-spaces real-numbers
edited Nov 10 '18 at 9:16
Christian Blatter
172k7113326
asked Nov 10 '18 at 8:53
GimgimGimgim
1629
I am stuck with a question that the constant $3$ in the Vitali covering lemma can not be replaced by any positive constant less than that in the finite case.
Observe that this question is different from the question of why $3$ is considered as bad constant because I am not considering infinite case here.
For example, I was thinking in real line and the argument can be generalised in the higher dimension; if I have $(0,4)$ and constant is $c<3$ then taking the interval $(8,4+frac{3-c}{2})$ or anything would not give me my answer. So I was doubting the validity of the statement in a metric space but I don't know whether I am missing anything or not!!
real-analysis general-topology measure-theory metric-spaces real-numbers
real-analysis general-topology measure-theory metric-spaces real-numbers
edited Nov 10 '18 at 9:16
Christian Blatter
172k7113326
asked Nov 10 '18 at 8:53
GimgimGimgim
1629
edited Nov 10 '18 at 9:16
Christian Blatter
172k7113326
asked Nov 10 '18 at 8:53
GimgimGimgim
1629
edited Nov 10 '18 at 9:16
Christian Blatter
172k7113326
edited Nov 10 '18 at 9:16
Christian Blatter
172k7113326
edited Nov 10 '18 at 9:16
Christian Blatter
172k7113326
172k7113326
asked Nov 10 '18 at 8:53
GimgimGimgim
1629
asked Nov 10 '18 at 8:53
GimgimGimgim
1629
asked Nov 10 '18 at 8:53
GimgimGimgim
1629
1629
add a comment |
add a comment |
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You idea is good. Even more simple: a collection consisting of two closed touching balls of equal radius shows that $3$ cannot be replaced by a smaller number.
answered Dec 5 '18 at 4:12
Alex RavskyAlex Ravsky
39.4k32181
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
oldest
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active
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votes
You idea is good. Even more simple: a collection consisting of two closed touching balls of equal radius shows that $3$ cannot be replaced by a smaller number.
answered Dec 5 '18 at 4:12
Alex RavskyAlex Ravsky
39.4k32181
add a comment |
You idea is good. Even more simple: a collection consisting of two closed touching balls of equal radius shows that $3$ cannot be replaced by a smaller number.
answered Dec 5 '18 at 4:12
Alex RavskyAlex Ravsky
39.4k32181
add a comment |
You idea is good. Even more simple: a collection consisting of two closed touching balls of equal radius shows that $3$ cannot be replaced by a smaller number.
answered Dec 5 '18 at 4:12
Alex RavskyAlex Ravsky
39.4k32181
You idea is good. Even more simple: a collection consisting of two closed touching balls of equal radius shows that $3$ cannot be replaced by a smaller number.
answered Dec 5 '18 at 4:12
Alex RavskyAlex Ravsky
39.4k32181
answered Dec 5 '18 at 4:12
Alex RavskyAlex Ravsky
39.4k32181
answered Dec 5 '18 at 4:12
Alex RavskyAlex Ravsky
39.4k32181
answered Dec 5 '18 at 4:12
Alex RavskyAlex Ravsky
39.4k32181
39.4k32181
add a comment |
add a comment |
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