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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
asked Aug 3 at 20:15
Walter Simon
698
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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
asked Aug 3 at 20:15
Walter Simon
698
add a comment |Â
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
asked Aug 3 at 20:15
Walter Simon
698
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
asked Aug 3 at 20:15
Walter Simon
698
edited 2 days ago
asked Aug 3 at 20:15
Walter Simon
698
asked Aug 3 at 20:15
Walter Simon
698
asked Aug 3 at 20:15
Walter Simon
698
698
add a comment |Â
add a comment |Â
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