# Questions tagged [convex-cone]

This tag should be used with convex cones defined as usual within the context of Euclidean spaces or topological vector spaces, and their applications within geometry and topology, optimization, combinatorics and any other related area of mathematics using the classic definition of convex cone.

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### If ${\rm ker}T\cap C_\infty=\{0\}$, prove that $T(C_\infty)=T(C)_\infty$

Let $\mathbb{E_1,E_2}$ be two finite dimension Euclidean spaces. $T\in\mathcal{L}(\mathbb{E_1,E_2})$ is a linear map. Given a closed set $C\subset\mathbb{E_1}$. Let $C_\infty$ denote the asymptotic ...
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### Does every pointed polyhedral cone in $\mathbb{R}^n$ generated by its $n+1$ extreme rays contain two vectors whose sum is in the interior?

This might be a stupid question, but I have been thinking and googling for some time now and I still cannot seem to find a response. I am interested in how many extreme rays a finitely generated (...
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### "Slicing off" the cone by a linear inequality leaves only vectors generated by the "remaining" cone generators (Lemma 1 from Gleeson & Ryan 1990)

Let $Ax \le b$ be a infeasible set of linear inequalities. Gleeson & Ryan (1990) derive a polyhedron with vertices being in a one-to-one mapping with the set of irreducible inconsistent subsets. ...
1 vote
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### Show that $S^{\star} + C$ is a finitely generated cone whose generators span $\mathbb{R}^m$

Let $C = \{A \lambda \, | \, \lambda \geq 0\}$ be a finitely generated cone for some $A \in \mathbb{R}^{m \times n}$. Suppose the columns of $A$ do not span $\mathbb{R}^m$. Let $S$ denote the column ...
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### Connected components of complement of double napped cone

How may connected components does the complement of the conic surface (extending on both sides) have in the three dimensional space? I think the connected components is two, but am not convinced about ...
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### Solid Angles Beyond Dimension Three

There is a theorem by Ribando (Measuring Solid Angles Beyond Dimension Three, Discrete Comput Geom 36:479–487 (2006)) https://link.springer.com/content/pdf/10.1007/s00454-006-1253-4.pdf?pdf=button: ...
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### convex polyhedral cone interpretation

According to my text, a convex polyhedral cone in $N_{\mathbb{R}}$, is a set of the form $\sigma = \big \lbrace \: a_1n_1+\cdots +a_rn_r \mid a_i\in \mathbb{R}_{\geq 0} \: \big \rbrace$ generated by a ...
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### Convex optimization problem not expressible as a conic program

I've been reading Boyd & Vandenberghe and it says that conic programming is a subclass of convex optimization. I haven't been able to find an example of a convex optimization problem that I cannot ...
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### Open discrete cones covering cover sphere?

This seems to be true, but I can't prove it. Define a cone to be $K\subset \mathbb{R}^n$ such that if $x\in K$ iff $\lambda x\in K$ for all $\lambda>0$. A discrete open cone $K'$ is the ...
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### A closed, convex set has a recession direction if and only if it is unbounded.

Let $C\subseteq \Bbb R^n$ be a nonempty closed, convex set. Prove that $C$ has a recession direction if and only if $C$ is unbounded. If $C$ has a recession direction then trivially it is unbounded. ...
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### Prove that the angle between an outer support vector and a unit vector of a cone is minimized on its extreme ray

Let $C$ be a closed convex cone and $\nu$ be an outer support vector, i.e. $\langle\nu, x\rangle\le 0$ for all $x\in C$. Assume $\nu$ is maximized uniquely at some point $y\in C$. I am trying to show ...
1 vote
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### Is half-space a convex cone?

I understand from Show that halfspace is not affine. that halfspace is not an affine set. Would like to ask is half-space a convex cone? How do you prove it mathematically? I am unsure if the ...
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