A conjecture concerning symmetric convex setsThe logic of convex setsPolynomial roots and convexityTo give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector functionBlow-Up for Semi-Linear Wave Equationscovering convex sets by round ballsOn an Inequality of Lars HörmanderUnderstanding the geometry of $H_n=x_i in [-N,N]:sum_i=1^n x_i = 0$Convex Hull of Outer Products of (Normalised) Nonnegative VectorsRandom Two-Player Asymmetric GameUnderstanding Polya, Hardy and Littlewood's definition of similarly-ordered

A conjecture concerning symmetric convex sets


The logic of convex setsPolynomial roots and convexityTo give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector functionBlow-Up for Semi-Linear Wave Equationscovering convex sets by round ballsOn an Inequality of Lars HörmanderUnderstanding the geometry of $H_n=x_i in [-N,N]:sum_i=1^n x_i = 0$Convex Hull of Outer Products of (Normalised) Nonnegative VectorsRandom Two-Player Asymmetric GameUnderstanding Polya, Hardy and Littlewood's definition of similarly-ordered













1












$begingroup$


 Question:



Let's suppose that $S subset mathbbR^n$ is convex and symmetric so:



beginequation
x in S iff -x in S tag1
endequation



Now, if we define the radius of $S$ as $R$ such that:



beginequation
R = sup_x in S lVert x rVert tag2
endequation



and use (2) to define:



beginequation
V = x in S: lVert x rVert = R tag3
endequation



then I conjecture that:



beginequation
S = textconv(V) tag*
endequation



I have worked out special cases of this problem within the context of high-dimensional probability but I suspect that it's generally true.



Might there be a theorem which guarantees this result?



Special case:



  1. As some people are voting to close this question I'd like to share my intuition about a special case as I think it might clarify my perspective.


  2. I was thinking in particular about symmetric convex polytopes and my intuition was that all symmetric convex polytopes in $mathbbR^n$ whose vertex set equalled $V$ in (3) were regular polytopes.


 Remark:



I consulted several texts on convexity in high dimensions and couldn't find an answer to this question. For this reason I decided to ask the question here.






reference-request real-analysis convex-polytopes convex-geometry







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    It's even worse than in the answers given below: unless $S$ is assumed to be closed, $V$ can well be empty!
    $endgroup$
    – Mateusz Kwaśnicki
    8 hours ago










  • $begingroup$
    I implicitly assumed we want to also assume that $S$ is closed and bounded (otherwise we can get $R=infty$ or $V=varnothing$).
    $endgroup$
    – user19871987
    6 hours ago















1












$begingroup$


 Question:



Let's suppose that $S subset mathbbR^n$ is convex and symmetric so:



beginequation
x in S iff -x in S tag1
endequation



Now, if we define the radius of $S$ as $R$ such that:



beginequation
R = sup_x in S lVert x rVert tag2
endequation



and use (2) to define:



beginequation
V = x in S: lVert x rVert = R tag3
endequation



then I conjecture that:



beginequation
S = textconv(V) tag*
endequation



I have worked out special cases of this problem within the context of high-dimensional probability but I suspect that it's generally true.



Might there be a theorem which guarantees this result?



Special case:



  1. As some people are voting to close this question I'd like to share my intuition about a special case as I think it might clarify my perspective.


  2. I was thinking in particular about symmetric convex polytopes and my intuition was that all symmetric convex polytopes in $mathbbR^n$ whose vertex set equalled $V$ in (3) were regular polytopes.


 Remark:



I consulted several texts on convexity in high dimensions and couldn't find an answer to this question. For this reason I decided to ask the question here.






reference-request real-analysis convex-polytopes convex-geometry







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    It's even worse than in the answers given below: unless $S$ is assumed to be closed, $V$ can well be empty!
    $endgroup$
    – Mateusz Kwaśnicki
    8 hours ago










  • $begingroup$
    I implicitly assumed we want to also assume that $S$ is closed and bounded (otherwise we can get $R=infty$ or $V=varnothing$).
    $endgroup$
    – user19871987
    6 hours ago













1












1








1





$begingroup$


 Question:



Let's suppose that $S subset mathbbR^n$ is convex and symmetric so:



beginequation
x in S iff -x in S tag1
endequation



Now, if we define the radius of $S$ as $R$ such that:



beginequation
R = sup_x in S lVert x rVert tag2
endequation



and use (2) to define:



beginequation
V = x in S: lVert x rVert = R tag3
endequation



then I conjecture that:



beginequation
S = textconv(V) tag*
endequation



I have worked out special cases of this problem within the context of high-dimensional probability but I suspect that it's generally true.



Might there be a theorem which guarantees this result?



Special case:



  1. As some people are voting to close this question I'd like to share my intuition about a special case as I think it might clarify my perspective.


  2. I was thinking in particular about symmetric convex polytopes and my intuition was that all symmetric convex polytopes in $mathbbR^n$ whose vertex set equalled $V$ in (3) were regular polytopes.


 Remark:



I consulted several texts on convexity in high dimensions and couldn't find an answer to this question. For this reason I decided to ask the question here.






reference-request real-analysis convex-polytopes convex-geometry







share|cite|improve this question











$endgroup$




 Question:



Let's suppose that $S subset mathbbR^n$ is convex and symmetric so:



beginequation
x in S iff -x in S tag1
endequation



Now, if we define the radius of $S$ as $R$ such that:



beginequation
R = sup_x in S lVert x rVert tag2
endequation



and use (2) to define:



beginequation
V = x in S: lVert x rVert = R tag3
endequation



then I conjecture that:



beginequation
S = textconv(V) tag*
endequation



I have worked out special cases of this problem within the context of high-dimensional probability but I suspect that it's generally true.



Might there be a theorem which guarantees this result?



Special case:



  1. As some people are voting to close this question I'd like to share my intuition about a special case as I think it might clarify my perspective.


  2. I was thinking in particular about symmetric convex polytopes and my intuition was that all symmetric convex polytopes in $mathbbR^n$ whose vertex set equalled $V$ in (3) were regular polytopes.


 Remark:



I consulted several texts on convexity in high dimensions and couldn't find an answer to this question. For this reason I decided to ask the question here.






reference-request real-analysis convex-polytopes convex-geometry




reference-request real-analysis convex-polytopes convex-geometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 1 hour ago







Aidan Rocke

















asked 10 hours ago









Aidan RockeAidan Rocke

515316




515316







  • 1




    $begingroup$
    It's even worse than in the answers given below: unless $S$ is assumed to be closed, $V$ can well be empty!
    $endgroup$
    – Mateusz Kwaśnicki
    8 hours ago










  • $begingroup$
    I implicitly assumed we want to also assume that $S$ is closed and bounded (otherwise we can get $R=infty$ or $V=varnothing$).
    $endgroup$
    – user19871987
    6 hours ago












  • 1




    $begingroup$
    It's even worse than in the answers given below: unless $S$ is assumed to be closed, $V$ can well be empty!
    $endgroup$
    – Mateusz Kwaśnicki
    8 hours ago










  • $begingroup$
    I implicitly assumed we want to also assume that $S$ is closed and bounded (otherwise we can get $R=infty$ or $V=varnothing$).
    $endgroup$
    – user19871987
    6 hours ago







1




1




$begingroup$
It's even worse than in the answers given below: unless $S$ is assumed to be closed, $V$ can well be empty!
$endgroup$
– Mateusz Kwaśnicki
8 hours ago




$begingroup$
It's even worse than in the answers given below: unless $S$ is assumed to be closed, $V$ can well be empty!
$endgroup$
– Mateusz Kwaśnicki
8 hours ago












$begingroup$
I implicitly assumed we want to also assume that $S$ is closed and bounded (otherwise we can get $R=infty$ or $V=varnothing$).
$endgroup$
– user19871987
6 hours ago




$begingroup$
I implicitly assumed we want to also assume that $S$ is closed and bounded (otherwise we can get $R=infty$ or $V=varnothing$).
$endgroup$
– user19871987
6 hours ago










2 Answers
2






active

oldest

votes


















4












$begingroup$

Consider the set
$$S:= (x,y) in mathbb R^2 : x^2+4y^2 leq 4 $$



This is convex and symmetric, and $R=2$.



But $V= (2,0), (-2,0) $ and $mboxconv(V)= (x, 0) : -2 leq x leq 2 neq S$.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Has this obstruction anything to do with the fact that the isometry group of S is of order greater than 2?
    $endgroup$
    – Sylvain JULIEN
    9 hours ago


















4












$begingroup$

Take any convex, symmetric, bounded set $T$ in $mathbbR^n$. Choose any point $pin mathbbR^n$ such that $|p|>sup_xin T|x|$ and let $S$ be the convex hull of $Tcup pm p$. This set is convex and symmetric, $V=pm p$, and the convex hull of $V$ is just the line segment connecting $p$ and $-p$. It is easy to construct counterexmples in this way, for example if the original set $T$ has non-empty interior.






share|cite|improve this answer









$endgroup$













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    2 Answers
    2






    active

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    2 Answers
    2






    active

    oldest

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    active

    oldest

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    active

    oldest

    votes









    4












    $begingroup$

    Consider the set
    $$S:= (x,y) in mathbb R^2 : x^2+4y^2 leq 4 $$



    This is convex and symmetric, and $R=2$.



    But $V= (2,0), (-2,0) $ and $mboxconv(V)= (x, 0) : -2 leq x leq 2 neq S$.






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Has this obstruction anything to do with the fact that the isometry group of S is of order greater than 2?
      $endgroup$
      – Sylvain JULIEN
      9 hours ago















    4












    $begingroup$

    Consider the set
    $$S:= (x,y) in mathbb R^2 : x^2+4y^2 leq 4 $$



    This is convex and symmetric, and $R=2$.



    But $V= (2,0), (-2,0) $ and $mboxconv(V)= (x, 0) : -2 leq x leq 2 neq S$.






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Has this obstruction anything to do with the fact that the isometry group of S is of order greater than 2?
      $endgroup$
      – Sylvain JULIEN
      9 hours ago













    4












    4








    4





    $begingroup$

    Consider the set
    $$S:= (x,y) in mathbb R^2 : x^2+4y^2 leq 4 $$



    This is convex and symmetric, and $R=2$.



    But $V= (2,0), (-2,0) $ and $mboxconv(V)= (x, 0) : -2 leq x leq 2 neq S$.






    share|cite|improve this answer









    $endgroup$



    Consider the set
    $$S:= (x,y) in mathbb R^2 : x^2+4y^2 leq 4 $$



    This is convex and symmetric, and $R=2$.



    But $V= (2,0), (-2,0) $ and $mboxconv(V)= (x, 0) : -2 leq x leq 2 neq S$.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 10 hours ago









    Nick SNick S

    872718




    872718











    • $begingroup$
      Has this obstruction anything to do with the fact that the isometry group of S is of order greater than 2?
      $endgroup$
      – Sylvain JULIEN
      9 hours ago
















    • $begingroup$
      Has this obstruction anything to do with the fact that the isometry group of S is of order greater than 2?
      $endgroup$
      – Sylvain JULIEN
      9 hours ago















    $begingroup$
    Has this obstruction anything to do with the fact that the isometry group of S is of order greater than 2?
    $endgroup$
    – Sylvain JULIEN
    9 hours ago




    $begingroup$
    Has this obstruction anything to do with the fact that the isometry group of S is of order greater than 2?
    $endgroup$
    – Sylvain JULIEN
    9 hours ago











    4












    $begingroup$

    Take any convex, symmetric, bounded set $T$ in $mathbbR^n$. Choose any point $pin mathbbR^n$ such that $|p|>sup_xin T|x|$ and let $S$ be the convex hull of $Tcup pm p$. This set is convex and symmetric, $V=pm p$, and the convex hull of $V$ is just the line segment connecting $p$ and $-p$. It is easy to construct counterexmples in this way, for example if the original set $T$ has non-empty interior.






    share|cite|improve this answer









    $endgroup$

















      4












      $begingroup$

      Take any convex, symmetric, bounded set $T$ in $mathbbR^n$. Choose any point $pin mathbbR^n$ such that $|p|>sup_xin T|x|$ and let $S$ be the convex hull of $Tcup pm p$. This set is convex and symmetric, $V=pm p$, and the convex hull of $V$ is just the line segment connecting $p$ and $-p$. It is easy to construct counterexmples in this way, for example if the original set $T$ has non-empty interior.






      share|cite|improve this answer









      $endgroup$















        4












        4








        4





        $begingroup$

        Take any convex, symmetric, bounded set $T$ in $mathbbR^n$. Choose any point $pin mathbbR^n$ such that $|p|>sup_xin T|x|$ and let $S$ be the convex hull of $Tcup pm p$. This set is convex and symmetric, $V=pm p$, and the convex hull of $V$ is just the line segment connecting $p$ and $-p$. It is easy to construct counterexmples in this way, for example if the original set $T$ has non-empty interior.






        share|cite|improve this answer









        $endgroup$



        Take any convex, symmetric, bounded set $T$ in $mathbbR^n$. Choose any point $pin mathbbR^n$ such that $|p|>sup_xin T|x|$ and let $S$ be the convex hull of $Tcup pm p$. This set is convex and symmetric, $V=pm p$, and the convex hull of $V$ is just the line segment connecting $p$ and $-p$. It is easy to construct counterexmples in this way, for example if the original set $T$ has non-empty interior.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 8 hours ago









        user19871987user19871987

        56739




        56739



























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