Supporting hyperplanes for closed convex cones
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Well I've been working on it for 3 days without results, it is not a question from any textbook and seems like it is not an important issue. But I really want to know the answers.
The definition of convex cone: C is a convex cone $Leftrightarrow forall x,yin C,foralltheta_1,theta_2geq0$, we have $theta_1 x+theta_2 yin C$ .
Question: suppose C is a non-trivial closed convex cone in n-dimensional Euclidean space,that means $Csubset R^n$, $C=cl(C)$, and C is neither a half-space nor a whole space, the question is:
1 Whether there exists a vector $ain R^n$ satisfies $forall xin Cbackslash{{0}}$, $a'x>0$.
2 Particularly, set $a= argmaxlimits_{|u|=1} minlimits_{xin C,|x|=1}{u'x}$, is the inequality above still correct? Prove it or give a counterexample.
optimization convex-cone
asked Jan 11 at 7:08
MorckMorck
11
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add a comment |
$begingroup$
Well I've been working on it for 3 days without results, it is not a question from any textbook and seems like it is not an important issue. But I really want to know the answers.
The definition of convex cone: C is a convex cone $Leftrightarrow forall x,yin C,foralltheta_1,theta_2geq0$, we have $theta_1 x+theta_2 yin C$ .
Question: suppose C is a non-trivial closed convex cone in n-dimensional Euclidean space,that means $Csubset R^n$, $C=cl(C)$, and C is neither a half-space nor a whole space, the question is:
1 Whether there exists a vector $ain R^n$ satisfies $forall xin Cbackslash{{0}}$, $a'x>0$.
2 Particularly, set $a= argmaxlimits_{|u|=1} minlimits_{xin C,|x|=1}{u'x}$, is the inequality above still correct? Prove it or give a counterexample.
optimization convex-cone
asked Jan 11 at 7:08
MorckMorck
11
$endgroup$
add a comment |
$begingroup$
Well I've been working on it for 3 days without results, it is not a question from any textbook and seems like it is not an important issue. But I really want to know the answers.
The definition of convex cone: C is a convex cone $Leftrightarrow forall x,yin C,foralltheta_1,theta_2geq0$, we have $theta_1 x+theta_2 yin C$ .
Question: suppose C is a non-trivial closed convex cone in n-dimensional Euclidean space,that means $Csubset R^n$, $C=cl(C)$, and C is neither a half-space nor a whole space, the question is:
1 Whether there exists a vector $ain R^n$ satisfies $forall xin Cbackslash{{0}}$, $a'x>0$.
2 Particularly, set $a= argmaxlimits_{|u|=1} minlimits_{xin C,|x|=1}{u'x}$, is the inequality above still correct? Prove it or give a counterexample.
optimization convex-cone
asked Jan 11 at 7:08
MorckMorck
11
$endgroup$
Well I've been working on it for 3 days without results, it is not a question from any textbook and seems like it is not an important issue. But I really want to know the answers.
The definition of convex cone: C is a convex cone $Leftrightarrow forall x,yin C,foralltheta_1,theta_2geq0$, we have $theta_1 x+theta_2 yin C$ .
Question: suppose C is a non-trivial closed convex cone in n-dimensional Euclidean space,that means $Csubset R^n$, $C=cl(C)$, and C is neither a half-space nor a whole space, the question is:
1 Whether there exists a vector $ain R^n$ satisfies $forall xin Cbackslash{{0}}$, $a'x>0$.
2 Particularly, set $a= argmaxlimits_{|u|=1} minlimits_{xin C,|x|=1}{u'x}$, is the inequality above still correct? Prove it or give a counterexample.
optimization convex-cone
optimization convex-cone
asked Jan 11 at 7:08
MorckMorck
11
asked Jan 11 at 7:08
MorckMorck
11
asked Jan 11 at 7:08
MorckMorck
11
asked Jan 11 at 7:08
MorckMorck
11
asked Jan 11 at 7:08
MorckMorck
11
11
add a comment |
add a comment |
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