Maximum of a linear function in a set that is convex but not compact

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Let $f: Vto mathbbR$ be a linear function defined on some real vector space $V$.



It is known that, if $U$ is a convex and compact subset of $V$, then $f$ attains its maximum at some extreme point of $U$.



Suppose $U$ is a convex subset of $V$, but not necessarily compact.
In this case, it is possible that $f$ does not attain its maximum in $V$.
What can be said about the maximum point of $f$ in this case? Is it true that $f$ attains its maximum in a boundary point of $V$?




real-analysis convex-analysis maxima-minima




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  • @MeesdeVries OP is considering linear functions.
    – Kavi Rama Murthy
    Jul 27 at 8:00














up vote
0
down vote

favorite












Let $f: Vto mathbbR$ be a linear function defined on some real vector space $V$.



It is known that, if $U$ is a convex and compact subset of $V$, then $f$ attains its maximum at some extreme point of $U$.



Suppose $U$ is a convex subset of $V$, but not necessarily compact.
In this case, it is possible that $f$ does not attain its maximum in $V$.
What can be said about the maximum point of $f$ in this case? Is it true that $f$ attains its maximum in a boundary point of $V$?




real-analysis convex-analysis maxima-minima




share|cite|improve this question



















  • @MeesdeVries OP is considering linear functions.
    – Kavi Rama Murthy
    Jul 27 at 8:00












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $f: Vto mathbbR$ be a linear function defined on some real vector space $V$.



It is known that, if $U$ is a convex and compact subset of $V$, then $f$ attains its maximum at some extreme point of $U$.



Suppose $U$ is a convex subset of $V$, but not necessarily compact.
In this case, it is possible that $f$ does not attain its maximum in $V$.
What can be said about the maximum point of $f$ in this case? Is it true that $f$ attains its maximum in a boundary point of $V$?




real-analysis convex-analysis maxima-minima




share|cite|improve this question











Let $f: Vto mathbbR$ be a linear function defined on some real vector space $V$.



It is known that, if $U$ is a convex and compact subset of $V$, then $f$ attains its maximum at some extreme point of $U$.



Suppose $U$ is a convex subset of $V$, but not necessarily compact.
In this case, it is possible that $f$ does not attain its maximum in $V$.
What can be said about the maximum point of $f$ in this case? Is it true that $f$ attains its maximum in a boundary point of $V$?





real-analysis convex-analysis maxima-minima





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asked Jul 27 at 7:32









Erel Segal-Halevi

4,10811757




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  • @MeesdeVries OP is considering linear functions.
    – Kavi Rama Murthy
    Jul 27 at 8:00
















  • @MeesdeVries OP is considering linear functions.
    – Kavi Rama Murthy
    Jul 27 at 8:00















@MeesdeVries OP is considering linear functions.
– Kavi Rama Murthy
Jul 27 at 8:00




@MeesdeVries OP is considering linear functions.
– Kavi Rama Murthy
Jul 27 at 8:00










1 Answer
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If $U$ is not compact, either its closure $bar U$ is bounded, then it is compact and convex (in which case you are right and the maximum will be attained on the boundary if it is not attained in the interior) or $bar U$ is unbounded. In the latter case it is possible that there is no maximum at all, for example for $f(x)=x$ and $U=mathbbR=V$.






share|cite|improve this answer





















  • What does it mean for a subset of a general vector space to be "bounded" - is it bounded in this sense en.wikipedia.org/wiki/Bounded_set_(topological_vector_space) ?
    – Erel Segal-Halevi
    Jul 27 at 7:58










  • Oh, I didn't pay enough attention. I was thinking of finite dimensional vector spaces only.
    – Kusma
    Jul 27 at 9:10










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If $U$ is not compact, either its closure $bar U$ is bounded, then it is compact and convex (in which case you are right and the maximum will be attained on the boundary if it is not attained in the interior) or $bar U$ is unbounded. In the latter case it is possible that there is no maximum at all, for example for $f(x)=x$ and $U=mathbbR=V$.






share|cite|improve this answer





















  • What does it mean for a subset of a general vector space to be "bounded" - is it bounded in this sense en.wikipedia.org/wiki/Bounded_set_(topological_vector_space) ?
    – Erel Segal-Halevi
    Jul 27 at 7:58










  • Oh, I didn't pay enough attention. I was thinking of finite dimensional vector spaces only.
    – Kusma
    Jul 27 at 9:10














up vote
0
down vote













If $U$ is not compact, either its closure $bar U$ is bounded, then it is compact and convex (in which case you are right and the maximum will be attained on the boundary if it is not attained in the interior) or $bar U$ is unbounded. In the latter case it is possible that there is no maximum at all, for example for $f(x)=x$ and $U=mathbbR=V$.






share|cite|improve this answer





















  • What does it mean for a subset of a general vector space to be "bounded" - is it bounded in this sense en.wikipedia.org/wiki/Bounded_set_(topological_vector_space) ?
    – Erel Segal-Halevi
    Jul 27 at 7:58










  • Oh, I didn't pay enough attention. I was thinking of finite dimensional vector spaces only.
    – Kusma
    Jul 27 at 9:10












up vote
0
down vote










up vote
0
down vote









If $U$ is not compact, either its closure $bar U$ is bounded, then it is compact and convex (in which case you are right and the maximum will be attained on the boundary if it is not attained in the interior) or $bar U$ is unbounded. In the latter case it is possible that there is no maximum at all, for example for $f(x)=x$ and $U=mathbbR=V$.






share|cite|improve this answer













If $U$ is not compact, either its closure $bar U$ is bounded, then it is compact and convex (in which case you are right and the maximum will be attained on the boundary if it is not attained in the interior) or $bar U$ is unbounded. In the latter case it is possible that there is no maximum at all, for example for $f(x)=x$ and $U=mathbbR=V$.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 27 at 7:45









Kusma

1,097111




1,097111











  • What does it mean for a subset of a general vector space to be "bounded" - is it bounded in this sense en.wikipedia.org/wiki/Bounded_set_(topological_vector_space) ?
    – Erel Segal-Halevi
    Jul 27 at 7:58










  • Oh, I didn't pay enough attention. I was thinking of finite dimensional vector spaces only.
    – Kusma
    Jul 27 at 9:10
















  • What does it mean for a subset of a general vector space to be "bounded" - is it bounded in this sense en.wikipedia.org/wiki/Bounded_set_(topological_vector_space) ?
    – Erel Segal-Halevi
    Jul 27 at 7:58










  • Oh, I didn't pay enough attention. I was thinking of finite dimensional vector spaces only.
    – Kusma
    Jul 27 at 9:10















What does it mean for a subset of a general vector space to be "bounded" - is it bounded in this sense en.wikipedia.org/wiki/Bounded_set_(topological_vector_space) ?
– Erel Segal-Halevi
Jul 27 at 7:58




What does it mean for a subset of a general vector space to be "bounded" - is it bounded in this sense en.wikipedia.org/wiki/Bounded_set_(topological_vector_space) ?
– Erel Segal-Halevi
Jul 27 at 7:58












Oh, I didn't pay enough attention. I was thinking of finite dimensional vector spaces only.
– Kusma
Jul 27 at 9:10




Oh, I didn't pay enough attention. I was thinking of finite dimensional vector spaces only.
– Kusma
Jul 27 at 9:10












 

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