# 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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### Blow up of toric variety corresponds to subdivision of cone

Look at the lattice $N= \mathbb{Z}^3/\mathbb{Z}(1,1,2)$, let $u_0,u_1,u_2$ be the images of the standard basis elements of $\mathbb{Z}^3$ and consider the cone $\sigma = \text{Cone}(u_0,u_1)$. Then it ...
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### Rewrite as conic optimization problem

Can I rewrite the convex optimization problem \begin{align} &\min \,\, f(x_1,x_2) \\ &\text{s.t.} \, \, x_1^2 + x_2^2 \leq r \end{align} with $r \in \mathbb{R}$ as a conic optimization problem?...
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### Second Order Cone Program with Quadratic Objective

The standard form for a Second Order Cone Program (SOCP) \begin{equation} \begin{array}{c} \min _{x} f^{T} x \\ \left\|A_{i} x+b_{i}\right\|_{2} \leq c_{i}^{T} x+d_{i}, i=1, \ldots, m \end{array} \end{...
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### farkas-minkowski theorem specific exmaple

I was reading about farkas-minkowski theorem, which basically said a convex cone is polyhedral iff it's finitely generated. The theorem make sense, but when I played with some example I met the ...
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### Group actions leaving the positive orthant invariant

Let $\Bbb R^n_{++} = \{x\in \Bbb R^n\mid x_1,\ldots,x_n> 0\}$ be the positive orthant. What are group actions which leaves $\Bbb R^n_{++}$ invariant? I think I have already found these ones: The ...
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### Difference of positive semi-definite matrices

If we have $S$ positive semi-definite matrices $A_1,\dots, A_S$ then what is the largest matrix positive semi definite matrix C such that $A_s -C$ is also psd for all $s=1,\dotsc,S$? By largest I mean ...
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### Interior angle of a polyhedral cone

What is the angle subtended by a polyhedral cone $\{\pmb{\theta}\in\mathbb{R}^{m}:A\pmb{\theta}\ge\pmb{0}\}$ at its vertex (the origin) where $A$ is a full-rank matrix ? Also what is the solid angle ...
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### Convex cone and orthogonal question

Let $K$ be a closed convex cone in $\mathbb{R}^n$. Show that $K = (K \cap -K) + (K \cap (K \cap -K)^{\perp})$ The above holds if $K = -K$, because $K$ is then a subspace. Similarly, it holds if $K$ ...
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### Meaning of continuous family of cones.

While reading about Ricci flow from Andrews and Hopper's book, I came across this notion of a continuous family of cones in a vector space $V$. I didn't find the definition for this notion in the book ...
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### What is the name of the opposite of a convex cone?

A subset $C$ of a vector space is a cone if for any element $x$ of $C$ and for any non-negative scalar $\alpha$, $\alpha x\in C$. Let $C$ be a cone. When the sum of any two elements of $C$ is also ...
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### Cone of feasible directions and radial cone

I am trying to prove if $A$ is convex and $x^*\in A$, then $D(A,x^*)=cone(A-x^*)$, where $D(A,x^*)=\{ d\in \mathbb{R^n}| \exists \delta >0$ such that $x^* +td \in A, \forall t \in (0,\delta) \}$ ...
I'm trying to solve a problem involving cones and their properties. Considering a simplicial cone (with $n$ edges by definition), $K\subset\mathbb{R}^{n}$, the claim is that $K$ is always a pointed ...
### $[A^{-1}(\mathbb{R} ^m _{-})]^* = A^{T}(\mathbb{R} ^m _{+})$ polar cone
I am trying to show that $[A^{-1}(\mathbb{R} ^m _{-})]^* = A^{T}(\mathbb{R} ^m _{+})$, where $A$ is a matrix, $A \in \mathbb{R} ^{m\times n}$, and $[A^{-1}(\mathbb{R} ^m _{-})]^*$ means polar cone of \$...