The dual of a regular polyhedral cone is regular

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A rational polyhedral cone in $mathbbR^n$ is a set of the form
$$sigma=lambda_1x_1+dots+lambda_kx_kin mathbbR^nmid lambda_iin mathbbR_geq 0 ; ;forall , 1leq ileq k$$
for some $x_1,dots,x_kinmathbbZ^n$. If the set $x_1,dots,x_k$ can be extended to a basis of $mathbbZ^n$ we say that $sigma$ is a regular (or sometimes smooth) cone.



The dual cone of $sigma$ is the set defined as
$$sigma^vee=yinmathbbR^n mid langle x,yranglegeq 0, ; forall ,xin sigma$$
and it is also a rational polyhedral cone.



How can I show that $sigma$ regular $implies$ $sigma^vee$ regular? If someone has a reference would be good also.




convex-analysis integer-lattices convex-geometry toric-geometry dual-cone




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    A rational polyhedral cone in $mathbbR^n$ is a set of the form
    $$sigma=lambda_1x_1+dots+lambda_kx_kin mathbbR^nmid lambda_iin mathbbR_geq 0 ; ;forall , 1leq ileq k$$
    for some $x_1,dots,x_kinmathbbZ^n$. If the set $x_1,dots,x_k$ can be extended to a basis of $mathbbZ^n$ we say that $sigma$ is a regular (or sometimes smooth) cone.



    The dual cone of $sigma$ is the set defined as
    $$sigma^vee=yinmathbbR^n mid langle x,yranglegeq 0, ; forall ,xin sigma$$
    and it is also a rational polyhedral cone.



    How can I show that $sigma$ regular $implies$ $sigma^vee$ regular? If someone has a reference would be good also.




    convex-analysis integer-lattices convex-geometry toric-geometry dual-cone




    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      A rational polyhedral cone in $mathbbR^n$ is a set of the form
      $$sigma=lambda_1x_1+dots+lambda_kx_kin mathbbR^nmid lambda_iin mathbbR_geq 0 ; ;forall , 1leq ileq k$$
      for some $x_1,dots,x_kinmathbbZ^n$. If the set $x_1,dots,x_k$ can be extended to a basis of $mathbbZ^n$ we say that $sigma$ is a regular (or sometimes smooth) cone.



      The dual cone of $sigma$ is the set defined as
      $$sigma^vee=yinmathbbR^n mid langle x,yranglegeq 0, ; forall ,xin sigma$$
      and it is also a rational polyhedral cone.



      How can I show that $sigma$ regular $implies$ $sigma^vee$ regular? If someone has a reference would be good also.




      convex-analysis integer-lattices convex-geometry toric-geometry dual-cone




      share|cite|improve this question













      A rational polyhedral cone in $mathbbR^n$ is a set of the form
      $$sigma=lambda_1x_1+dots+lambda_kx_kin mathbbR^nmid lambda_iin mathbbR_geq 0 ; ;forall , 1leq ileq k$$
      for some $x_1,dots,x_kinmathbbZ^n$. If the set $x_1,dots,x_k$ can be extended to a basis of $mathbbZ^n$ we say that $sigma$ is a regular (or sometimes smooth) cone.



      The dual cone of $sigma$ is the set defined as
      $$sigma^vee=yinmathbbR^n mid langle x,yranglegeq 0, ; forall ,xin sigma$$
      and it is also a rational polyhedral cone.



      How can I show that $sigma$ regular $implies$ $sigma^vee$ regular? If someone has a reference would be good also.





      convex-analysis integer-lattices convex-geometry toric-geometry dual-cone





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      asked Aug 3 at 20:15









      Walter Simon

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