Mixing
Normal of a supporting hyperplane contained in the normal cone of a vertex
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Given a vertex $v$ of a polytope P defined by the intersection of $n$ linearly independent hyperplanes with normals $v_1, ldots, v_n$ and a supporting hyperplane $c^Tx leq d$ that passes through $v$, is it true $c in Cone(v_1, ldots, v_n)$?
This seems like it should be obvious, but I cannot find a proof of it anywhere. Thanks for the help.
Intuition why it should be true: if the normal is outside of this cone, the supporting hyperplane should cut into P, contradiction that it is a supporting hyperplane.
Unfortunately I have no clue how to formalize this.
My attempt so far: Farkas lemma says that if $c$ is not in the cone then there exists $a in mathbb{R}^n$ such that $c^Ta<0$ and $v_i^Ta geq 0$. Maybe there is a way to scale $a$ so that it satisfies the hyperplanes defining the vertex, but falsifies the supporting hyperplane, thus contradicting that this hyerplane is supporting.
linear-algebra convex-analysis convex-geometry
asked Jan 31 at 0:00
Yugioh MishimaYugioh Mishima
112
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add a comment |
$begingroup$
Given a vertex $v$ of a polytope P defined by the intersection of $n$ linearly independent hyperplanes with normals $v_1, ldots, v_n$ and a supporting hyperplane $c^Tx leq d$ that passes through $v$, is it true $c in Cone(v_1, ldots, v_n)$?
This seems like it should be obvious, but I cannot find a proof of it anywhere. Thanks for the help.
Intuition why it should be true: if the normal is outside of this cone, the supporting hyperplane should cut into P, contradiction that it is a supporting hyperplane.
Unfortunately I have no clue how to formalize this.
My attempt so far: Farkas lemma says that if $c$ is not in the cone then there exists $a in mathbb{R}^n$ such that $c^Ta<0$ and $v_i^Ta geq 0$. Maybe there is a way to scale $a$ so that it satisfies the hyperplanes defining the vertex, but falsifies the supporting hyperplane, thus contradicting that this hyerplane is supporting.
linear-algebra convex-analysis convex-geometry
asked Jan 31 at 0:00
Yugioh MishimaYugioh Mishima
112
$endgroup$
add a comment |
$begingroup$
Given a vertex $v$ of a polytope P defined by the intersection of $n$ linearly independent hyperplanes with normals $v_1, ldots, v_n$ and a supporting hyperplane $c^Tx leq d$ that passes through $v$, is it true $c in Cone(v_1, ldots, v_n)$?
This seems like it should be obvious, but I cannot find a proof of it anywhere. Thanks for the help.
Intuition why it should be true: if the normal is outside of this cone, the supporting hyperplane should cut into P, contradiction that it is a supporting hyperplane.
Unfortunately I have no clue how to formalize this.
My attempt so far: Farkas lemma says that if $c$ is not in the cone then there exists $a in mathbb{R}^n$ such that $c^Ta<0$ and $v_i^Ta geq 0$. Maybe there is a way to scale $a$ so that it satisfies the hyperplanes defining the vertex, but falsifies the supporting hyperplane, thus contradicting that this hyerplane is supporting.
linear-algebra convex-analysis convex-geometry
asked Jan 31 at 0:00
Yugioh MishimaYugioh Mishima
112
$endgroup$
Given a vertex $v$ of a polytope P defined by the intersection of $n$ linearly independent hyperplanes with normals $v_1, ldots, v_n$ and a supporting hyperplane $c^Tx leq d$ that passes through $v$, is it true $c in Cone(v_1, ldots, v_n)$?
This seems like it should be obvious, but I cannot find a proof of it anywhere. Thanks for the help.
Intuition why it should be true: if the normal is outside of this cone, the supporting hyperplane should cut into P, contradiction that it is a supporting hyperplane.
Unfortunately I have no clue how to formalize this.
My attempt so far: Farkas lemma says that if $c$ is not in the cone then there exists $a in mathbb{R}^n$ such that $c^Ta<0$ and $v_i^Ta geq 0$. Maybe there is a way to scale $a$ so that it satisfies the hyperplanes defining the vertex, but falsifies the supporting hyperplane, thus contradicting that this hyerplane is supporting.
linear-algebra convex-analysis convex-geometry
linear-algebra convex-analysis convex-geometry
asked Jan 31 at 0:00
Yugioh MishimaYugioh Mishima
112
asked Jan 31 at 0:00
Yugioh MishimaYugioh Mishima
112
edited Jan 31 at 1:19
Yugioh Mishima
asked Jan 31 at 0:00
Yugioh MishimaYugioh Mishima
112
asked Jan 31 at 0:00
Yugioh MishimaYugioh Mishima
112
asked Jan 31 at 0:00
Yugioh MishimaYugioh Mishima
112
112
add a comment |
add a comment |
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