All Questions
6 questions
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Basis of the intersection of a cone and its dual.
Let $a_1,\dots, a_m$ be vectors in $\mathbb R^n$, let $A\in\mathbb R^{n\times m}$ whose columns are $a_i$s. The cone generated by $A$ (denoted $\operatorname{cone}A$) is $C=\{ Aw: 0\leq w \}$, its ...
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Questions related to Cones and Subspaces of Euclidean Space
Cone: A subset $ S \subseteq \mathbb{R}^n$ is a cone if $\alpha \geq 0 \implies \alpha S \subseteq S.$
Polar: A Polar $K^*$ of a cone $K$ is a closed convex cone such that
$$K^*=\{y \in \mathbb{R}^n \...
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A cone in $\mathbb{R}^n$ containing n linearly independent vectors has a non empty interior
I need a help with proving, that if a cone $K \subseteq \mathbb{R}^n$ contains $n$ linearly independent vectors, then the interior of $K$ is non empty.
Lets say $b_1,\dots,b_n \in K$ are the ...
- 323
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Prooving the property of a polyhedral cone
Let me introduce some definitions. Poly$(v_1, v_2,\ldots, v_k) = \{ x \in \mathbb{R}^n \ | \ (x, v_i) \geq 0 \ \ \forall i \}$ called a polyhedral cone. For any cone $C$, $ \ C^{\lor} = \{ x \in \...
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Prove that a cone C is full-dimensional if and only if its dual cone $C^*$ is pointed.
A cone with apex $0$ is said to be pointed if it does not contain any non-trivial subspace. Let C be a closed convex cone with apex $0$. Show that $C$ is full-dimensional if and only if its dual cone $...
user533068
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Does Gordan's Lemma hold in infinite dimensional vector spaces?
Gordan's Lemma: Let $A \in \mathbb{R}^{m \times n}$. Then exactly one of the following two statements is true:
There exists $x \in \mathbb{R}^n$ with $Ax > 0$, or
There exists nonzero $y ...
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