Mathematics Stack Exchange
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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.

A convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients

261 questions
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Is a convex cone generated by a bounded convex closed set containing the origin closed?

Consider a closed bounded convex set in the space of Lebesgue integrable functions L^P that contains the origin. Is a convex cone generated by the set closed?
Alexey Kushnir
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How to proof the intersection of two closed convex cones is a closed convex cone?

Let $C$ and $D$ be closed convex cones in $R^n$. I am trying to show that $C\cap D$ is a closed cone. I started with Take any point $x_1 \in C$ and $x_2 \in D$ with $\theta_1,\theta_2\geq0$ and $\theta_1+\theta_2=1$. After that I do not think I…
daultongray8
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Equivalent definitions of the second-order/Lorentz cone

I have come across two different definitions of the second-order/Lorentz cone. The first is the standard form where $t$ is a scalar and $\mathbf{y} \in \mathbb{R}^n$. $$ \mathcal{C}_1 = \bigg\{ \begin{bmatrix}\mathbf{y} \\ t\end{bmatrix} \in…
Srinivas Eswar
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Intersection of two conic hulls is equal to the conic hull of the intersection

since the above conjecture is wrong in general, I would like to know (and maybe prove) that the following Statement holds: Let $A,B$ be two closed, convex sets in $\mathbb{R}^n$ such that $A\cap B=\text{bd}(A)\cap \text{bd}(B)\neq\emptyset$ (where…
Kluyvert
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How to decide whether a given point is inside or outside a given cone?

Given a cone defined by peak point $(X_0,Y_0,Z_0)$, bottom point $(X_1,Y_1,Z_1)$ and radius $R$, how can I decide whether a given point $(X',Y'Z')$ is inside the cone? Cone is not parallel to $XY$ plane. Cone can be at any angles based on peak…
manoos
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Is the union of the complement set of a cone set and the origin also be a cone set?

We call the set $X$ a cone if $\alpha x\in X, \forall x\in X, \alpha\succeq 0$. Thinking the cases in $\mathbb{R}, \mathbb{R}^2$, and $\Bbb{R}^3$, if $X$ is a cone, $X^C\cup\{0\}$ is also a cone, isn't it? ex) $[0,\infty]^C\cup\{0\}=[-\infty,0]$ is…
Danny_Kim
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Are all pointed cones convex?

If a cone is pointed, does that imply it is convex? It feels like it is true, but I want to be sure, since I can't seem to find it outright stated anywhere. For a cone $K$, if $\forall x \neq 0 \in K$, $-x \notin K$, then the cone must be restricted…
MegaZeroX
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How to find voxels intersected with a cone?

I have a 3D cube with 128x128x128 voxels. A cone is passing through this 3D cube. How can I find which voxels of cube are inside cone?
manoos
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Question regarding an answer on dual of a dual cone

Regarding this answer. We can take $K=\{(0,0)\}\cup\{(a,b):~a>0,b>0\}$. We have $K^*=\{(a,b):~a\geq 0,b\geq 0\}$. Let us take $C=\{(0,0)\}\cup\{(a,b):~a\geq 0,b>0\}$ and notice $K \subset C$. But we do not have $K^{**}\subset C$ as stated in your…
Daniel Porumbel
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