Equivalence of defining neighborhood as an open set or as a closed set in a special case
Let continuous $f:Xtomathbb R$. $X$ is an interval of $mathbb R$.
Are the following two statements equivalent?
1) For almost every $xin X$ $exists$ open interval (neighborhood) $Ini x$ s.t. $f$ is either locally convex or locally concave on $I$. (i.e. only countable number of $xin X$ does not have such neighborhood).
$f$ is locally convex on $I$ if $forall x,yin I$, we have $f(lambda x+(1-lambda)y)leq f(lambda x)+f((1-lambda)y)$
(In another word, epi$f(I)$ is a convex set.)
2) $forall xin X$ $exists$ (non-singleton) closed interval $Ini x$ s.t. $f$ is either locally convex or locally concave over $I$.
If they are equivalent, then, in general, why people tend to use open set rather than closed set to define a neighborhood?
real-analysis general-topology analysis continuity convex-analysis
asked Dec 26 '18 at 23:00
High GPA
889419
|
show 4 more comments
Let continuous $f:Xtomathbb R$. $X$ is an interval of $mathbb R$.
Are the following two statements equivalent?
1) For almost every $xin X$ $exists$ open interval (neighborhood) $Ini x$ s.t. $f$ is either locally convex or locally concave on $I$. (i.e. only countable number of $xin X$ does not have such neighborhood).
$f$ is locally convex on $I$ if $forall x,yin I$, we have $f(lambda x+(1-lambda)y)leq f(lambda x)+f((1-lambda)y)$
(In another word, epi$f(I)$ is a convex set.)
2) $forall xin X$ $exists$ (non-singleton) closed interval $Ini x$ s.t. $f$ is either locally convex or locally concave over $I$.
If they are equivalent, then, in general, why people tend to use open set rather than closed set to define a neighborhood?
real-analysis general-topology analysis continuity convex-analysis
asked Dec 26 '18 at 23:00
High GPA
889419
What do you mean by "locally convex"? Just that $f$ is convex on the corresponding interval?
– 0x539
Dec 26 '18 at 23:19
@0x539 Thank you very much for the note. You are right. The question is clarified.
– High GPA
Dec 26 '18 at 23:41
What do you mean by "$X$ is an interval of $mathbb{R}$"? Can $X$ be any one of $[a,b]$, $[a,b)$, $(a,b)$ or $(a,b]$?
– user587192
Dec 26 '18 at 23:45
@user587192 How about this? For simplicity let's just consider the compact case at first.
– High GPA
Dec 26 '18 at 23:47
1
@mathworker21 The specific case will help me understand the situation. I agree with you overall. I will edit the question to be more specific.
– High GPA
Dec 27 '18 at 1:14
|
show 4 more comments
Let continuous $f:Xtomathbb R$. $X$ is an interval of $mathbb R$.
Are the following two statements equivalent?
1) For almost every $xin X$ $exists$ open interval (neighborhood) $Ini x$ s.t. $f$ is either locally convex or locally concave on $I$. (i.e. only countable number of $xin X$ does not have such neighborhood).
$f$ is locally convex on $I$ if $forall x,yin I$, we have $f(lambda x+(1-lambda)y)leq f(lambda x)+f((1-lambda)y)$
(In another word, epi$f(I)$ is a convex set.)
2) $forall xin X$ $exists$ (non-singleton) closed interval $Ini x$ s.t. $f$ is either locally convex or locally concave over $I$.
If they are equivalent, then, in general, why people tend to use open set rather than closed set to define a neighborhood?
real-analysis general-topology analysis continuity convex-analysis
asked Dec 26 '18 at 23:00
High GPA
889419
Let continuous $f:Xtomathbb R$. $X$ is an interval of $mathbb R$.
Are the following two statements equivalent?
1) For almost every $xin X$ $exists$ open interval (neighborhood) $Ini x$ s.t. $f$ is either locally convex or locally concave on $I$. (i.e. only countable number of $xin X$ does not have such neighborhood).
$f$ is locally convex on $I$ if $forall x,yin I$, we have $f(lambda x+(1-lambda)y)leq f(lambda x)+f((1-lambda)y)$
(In another word, epi$f(I)$ is a convex set.)
2) $forall xin X$ $exists$ (non-singleton) closed interval $Ini x$ s.t. $f$ is either locally convex or locally concave over $I$.
If they are equivalent, then, in general, why people tend to use open set rather than closed set to define a neighborhood?
real-analysis general-topology analysis continuity convex-analysis
real-analysis general-topology analysis continuity convex-analysis
asked Dec 26 '18 at 23:00
High GPA
889419
asked Dec 26 '18 at 23:00
High GPA
889419
edited Dec 27 '18 at 1:14
asked Dec 26 '18 at 23:00
High GPA
889419
asked Dec 26 '18 at 23:00
High GPA
889419
asked Dec 26 '18 at 23:00
High GPA
889419
889419
What do you mean by "locally convex"? Just that $f$ is convex on the corresponding interval?
– 0x539
Dec 26 '18 at 23:19
@0x539 Thank you very much for the note. You are right. The question is clarified.
– High GPA
Dec 26 '18 at 23:41
What do you mean by "$X$ is an interval of $mathbb{R}$"? Can $X$ be any one of $[a,b]$, $[a,b)$, $(a,b)$ or $(a,b]$?
– user587192
Dec 26 '18 at 23:45
@user587192 How about this? For simplicity let's just consider the compact case at first.
– High GPA
Dec 26 '18 at 23:47
1
@mathworker21 The specific case will help me understand the situation. I agree with you overall. I will edit the question to be more specific.
– High GPA
Dec 27 '18 at 1:14
|
show 4 more comments
What do you mean by "locally convex"? Just that $f$ is convex on the corresponding interval?
– 0x539
Dec 26 '18 at 23:19
@0x539 Thank you very much for the note. You are right. The question is clarified.
– High GPA
Dec 26 '18 at 23:41
What do you mean by "$X$ is an interval of $mathbb{R}$"? Can $X$ be any one of $[a,b]$, $[a,b)$, $(a,b)$ or $(a,b]$?
– user587192
Dec 26 '18 at 23:45
@user587192 How about this? For simplicity let's just consider the compact case at first.
– High GPA
Dec 26 '18 at 23:47
1
@mathworker21 The specific case will help me understand the situation. I agree with you overall. I will edit the question to be more specific.
– High GPA
Dec 27 '18 at 1:14
What do you mean by "locally convex"? Just that $f$ is convex on the corresponding interval?
– 0x539
Dec 26 '18 at 23:19
What do you mean by "locally convex"? Just that $f$ is convex on the corresponding interval?
– 0x539
Dec 26 '18 at 23:19
@0x539 Thank you very much for the note. You are right. The question is clarified.
– High GPA
Dec 26 '18 at 23:41
@0x539 Thank you very much for the note. You are right. The question is clarified.
– High GPA
Dec 26 '18 at 23:41
What do you mean by "$X$ is an interval of $mathbb{R}$"? Can $X$ be any one of $[a,b]$, $[a,b)$, $(a,b)$ or $(a,b]$?
– user587192
Dec 26 '18 at 23:45
What do you mean by "$X$ is an interval of $mathbb{R}$"? Can $X$ be any one of $[a,b]$, $[a,b)$, $(a,b)$ or $(a,b]$?
– user587192
Dec 26 '18 at 23:45
@user587192 How about this? For simplicity let's just consider the compact case at first.
– High GPA
Dec 26 '18 at 23:47
@user587192 How about this? For simplicity let's just consider the compact case at first.
– High GPA
Dec 26 '18 at 23:47
1
1
@mathworker21 The specific case will help me understand the situation. I agree with you overall. I will edit the question to be more specific.
– High GPA
Dec 27 '18 at 1:14
@mathworker21 The specific case will help me understand the situation. I agree with you overall. I will edit the question to be more specific.
– High GPA
Dec 27 '18 at 1:14
|
show 4 more comments
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What do you mean by "locally convex"? Just that $f$ is convex on the corresponding interval?
– 0x539
Dec 26 '18 at 23:19
@0x539 Thank you very much for the note. You are right. The question is clarified.
– High GPA
Dec 26 '18 at 23:41
What do you mean by "$X$ is an interval of $mathbb{R}$"? Can $X$ be any one of $[a,b]$, $[a,b)$, $(a,b)$ or $(a,b]$?
– user587192
Dec 26 '18 at 23:45
@user587192 How about this? For simplicity let's just consider the compact case at first.
– High GPA
Dec 26 '18 at 23:47
1
@mathworker21 The specific case will help me understand the situation. I agree with you overall. I will edit the question to be more specific.
– High GPA
Dec 27 '18 at 1:14