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.

Filter by
Sorted by
Tagged with
0
votes
0answers
13 views

dual problem of linear programming over convex cones

Let $C$ denote a convex cone in $\mathbb{R}^n$ and $c$ a vector. Consider the problem $$\min_{x \in C} x^Tc.$$ Suppose $C^\star$ is the convex dual cone of $C$, what is its dual problem in terms of $C^...
1
vote
0answers
21 views

Locally polyhedral and locally finitely generated cones

Let $V$ be a finite dimensional real vector space. The following definitions are taken from a paper. A closed convex cone $K \subset V$ is locally polyhedral at $v \in \partial K$ if there exists a ...
1
vote
1answer
18 views

Example of the equivalence of two desrciption of polyhedra

This is an extract from a linear optimization book: However, I do not understand why both of these conditions describe the same polyhedra? The linear equation results in a set $$\left\{\begin{bmatrix}...
2
votes
0answers
34 views

Hyperbolic Coxeter groups, Humphreys' book

Let $(W,S)$ be an irreducible Coxeter system with non-degenerate bilinear form $B$ on the Euclidean vector space $V$. The simple root attached to $s\in S$ is denoted by $\alpha_s\in V$. Let $\{\...
2
votes
0answers
43 views

Support function of convex cone

Let $K ⊆ \mathbb{R}^{d}$ be a non-empty, closed, convex cone. Consider the support function $\sigma_K(x):= \sup_{y \in K} \langle x, y \rangle$. This function describes the (signed) distances of ...
1
vote
1answer
27 views

Prove that normal cone of a closed convex set is different from the set has only element 0

Problem: Suppose that $E$ is Euclidean space. Given a closed convex set $C \subset E,$ the normal cone $N_C$ to $C$ at $x \in C$ is defined by $$N_C(x) := \left\{v \in E^{*}:\forall y \in C, \langle ...
1
vote
0answers
23 views

Cone described by quadratic form

Let $P,Q$ be two real symmetric matrices of dimension $n \times n$. Define two sets $A=\{x \in \mathbb{R}^n: x^T P x \le 0 \}$ and $B = \{ x \in \mathbb{R}^n: x^T Q x \le 0 \}$. If $A \subset B$, ...
1
vote
0answers
14 views

Intersection multiplicity on toric varieties using simplicial cones

In the book "Introduction to toric varieties" by William Fulton (p. 100) there is a formula for the intersection product of two subvarieties. Let $\tau,\sigma$ be cones in the fan $\Delta$, ...
3
votes
1answer
25 views

Linear maps preserving a revolution cone

Let $C$ be the cone of $R^{n+1}$ defined by $$ C=\lbrace x=(x_1,..,x_{n+1})|\quad \sum_1^n x_i^2 >x_{n+1}^2 \rbrace.$$ I am interested in the set $\mathcal{C}$ of $n\times n$ matrices $M$ which ...
9
votes
1answer
104 views

Volume of the intersection of the unit ball with a polyhedral cone

Given vectors $x_1,...,x_n\in\Bbb R^d$. The conic span of these vectors is $$\mathrm{cone}\{x_1,...,x_n\}:=\{\alpha_1 x_1+\cdots +\alpha_n x_n\mid \alpha_1,...,\alpha_n\ge 0\}.$$ Question: Is there a ...
5
votes
1answer
66 views

Examples of interesting cones in infinite-dimensional Hilbert spaces

Let $X$ be a real Hilbert space, and let $K$ be a closed convex cone. The dual cone is defined by $K^*=\{x^*\in X \mid (\forall k\in K)\, \langle x^*,k\rangle \geq 0 \}$. I am looking for some ...
0
votes
1answer
30 views

Show that $M=\{x\in \mathbb{R}^n: x^TQx\leq(a^Tx)^2, a^Tx\geq0\}$ is convex cone. [closed]

I want to show that $M=\{x\in \mathbb{R}^n: x^TQx\leq(a^Tx)^2, a^Tx\geq0\}$ where $Q$ is positive definite is a convex cone. I know a theorem that says $S\subseteq \mathbb{R}^n$ is convex cone if and ...
0
votes
1answer
43 views

Subdifferential of normal cone proof

Problem: I'm stuck showing the opposite direction. I wanna show that $N(x̄,Ω)=∂δΩ(x̄)$ subdifferential of normal cone, where $N(x̄,Ω)=\{g \in Rn | \langle g, z-x̄\rangle ≤ 0, ∀ z ∈ Ω\}$. My attempt: ...
0
votes
0answers
15 views

Correctness proof second order cone is solid.

I would like to prove that the second order cone $$\mathit{2}^{n} = \{(x,c) \in \mathbb{R}^{n} \times \mathbb{R}_{\geq 0} : ||x||_{2} \leq c \}$$ is solid (as part of a proof that it is a proper cone)...
0
votes
0answers
21 views

Find the edges of a particular convex polyhedral cone

Let $A \in \mathbb{R}^{m \times n}$ with $m > n$. Consider $q \in \mathbb{R}^m$ and the set $$\mathcal{C} = \{ q \in \mathbb{R}^m | q^T \cdot A = 0, q \succeq 0\}$$ I think this is the intersection ...
0
votes
1answer
29 views

Prove a cone $S=\{(x_1,x_2,x_3) \in \mathbb{R}^3 : x_1^2+x_2^2-x_3^2=0, x_3 \ge 0 \}$ is a convex set.

I have to prove a cone $$S=\{(x_1,x_2,x_3) \in \mathbb{R}^3 : x_1^2+x_2^2-x_3^2=0, x_3 \ge 0 \}$$ is a convex set. The definition of a convex set is known as "A set $C$ is convex if for $x,y \in ...
0
votes
0answers
4 views

Diameter of $S=(a+K)\cap (b-K)$, where $K$ is a an acute cone and $b\in a+K$.

Let $K$ be an acute cone in $\mathbb R^n$, i.e. if $u\in K$ then $tu\in K$ for all $t\geq 0$; and $\langle u, v\rangle\geq 0$ for all $u, v\in K$. Let $a, b$ be two points in $\mathbb R^n$ such that $...
3
votes
0answers
69 views

Prove that this set is a convex cone

Let $$S := \left\{ x \in \mathbb{R}^n : x^T Q x \leq (a^Tx)^2, a^T x \geq0 \right\}$$ where $Q \in \mathbb{S}_{++}^n$ and $a \in \mathbb{R}^n$. Show that $S$ is a convex cone. I am aware that for a ...
3
votes
0answers
40 views

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?
2
votes
1answer
40 views

Given $x \in {\rm int}(K)$ (covered by nondecreasing cones $\{K^r\}$), does there exist $r_0$ s.t. $x \in {\rm int}(K^{r_0})$?

Let $K\ (\subseteq \mathbb{R}^n)$ be a closed convex cone and $\{K^r\}_{r=0}^\infty$ be a family of closed convex cones satisfying $K^r \subseteq K^{r+1} \subseteq K\ (\forall r)$. Assume that ${\rm ...
2
votes
0answers
59 views

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 ...
0
votes
1answer
19 views

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?...
0
votes
0answers
33 views

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{...
0
votes
0answers
27 views

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 ...
1
vote
1answer
52 views

Intersection of a pointed cone and a hyperplane is a polytope?

Let $C = \text{cone}(u_1,\dots,u_m)$ for some $u_1,\dots,u_m \in \mathbb{R}^d \setminus \{\textbf{0}\}$ be a finitely generated pointed cone. Let $H_0 := \{x \in \mathbb{R}^d: \langle a,x \rangle = 0\}...
0
votes
0answers
36 views

Why is a strongly convex rational polyhedral cone generated by the ray generators?

In the book "Toric Varieties" by Cox, Little and Schenck, Lemma 1.2.15 says 'A strongly convex rational polyhedral cone is generated by the ray generators of its edges.', but they did not ...
1
vote
1answer
28 views

Example of an unbounded convex set with no recession direction?

Is it true that every unbounded convex set in $R^n$ has a recession direction? I think there is a counterexample, because I have not assumed that the convex set is closed. I have not been able to come ...
0
votes
0answers
41 views

Convexity of the root of a complicated term

I am currently trying to prove the convexity of the following function of x, and I unfortunately have problems finding a solution $$ \sqrt{\mathbf{x^{\top}}\mathbf{Q} \mathbf{x} + 2(\boldsymbol{\...
0
votes
1answer
52 views

Cone equations given its vertex and a sphere circumscribe in it

In a solved problem it appears the following equations describing the points in a cone with vertex $v=(a,b,c)$ wich circumscribe the sphere $x^2+y^2+z^2=1$ : $$\mathcal{C}: (ax+by+cz-1)^2=(x^2+y^2+z^2-...
2
votes
1answer
48 views

Does a cone in $\mathbb{R}^3$ admit a warped geometry?

Let $$C = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 \leq r^2 z^2 \text{ and } z > 0 \}$$ be a certain convex cone in $\mathbb{R}^3$, where $r > 0$. Let $D$ the the closed unit disk in $\mathbb{R}^...
0
votes
0answers
49 views

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 ...
5
votes
0answers
30 views

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 ...
0
votes
0answers
40 views

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 ...
1
vote
1answer
53 views

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{...
1
vote
1answer
222 views

Norm cone is a proper cone

For a finite vector space $H$ define the norm cone $K = \{ (x, \lambda) \in H \oplus \mathbb{R} : \lVert x \rVert \le \lambda \}$ where $\lVert x \rVert$ is some norm. There are endless lecture notes ...
0
votes
0answers
86 views

Computing the Chebyshev center of a V-representation polyhedral cone

I'd like to compute the Chebyshev center of a convex polyhedral cone (restricting the center to be inside the unit hypercube). Given a polyhedral cone in H-representation (linear inequalities $\mathbf{...
0
votes
0answers
41 views

Orthogonal Projections on Polyhedral Cones

Suppose there are two distinct tetrahedral cones in $\mathbb{R}^{3}$ with vertices at the origin. Each cone is formed by three rays emanating from the origin. (i) Is it possible to project (...
0
votes
0answers
57 views

Difference of two closed convex cone that does not contain (1,...,1)

Let $a_1, \dots, a_p$ be $p$ vectors in $\mathbb{R}^n$ and $K$ the closed convex cone generated by $(a_1, \dots, a_p)$, i.e. $$K = \left\{ \sum_{j=1}^p \lambda_i a_i | (\lambda_i) \in \mathbb{R}^p_+ \...
0
votes
1answer
34 views

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 $K$ ...
1
vote
0answers
24 views

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 ...
1
vote
0answers
92 views

Why is a Quadratic programming problem a Second-order cone programming?

I have seen that QP can be rewritten as a SOCP from several resources online. Why is this the case? More precisely, using relaxation, QP can be written as $$ \min_{x,t} c^Tx + t ~\text{ subject to } ...
0
votes
2answers
352 views

What is the distance between the centre of the sphere and the vertex of the cone?

A hollow right circular cone rests on a sphere as shown in the figure. The height of the cone is $4$ metres and the radius of the base is $1$ metre. The volume of the sphere is same as that of the ...
2
votes
2answers
98 views

Defining the Polar set

For a subset P of R^n (real numbers) the polar set is defined by: $$ P^*:= \{ y\in \Bbb R^n\mid y\cdot x \leq 1 \text{ for all } x\in P \}. $$ Can someone break the definition into plain english as ...
1
vote
0answers
37 views

Difference of closed convex cones

I have read the following claim The difference of two closed convex cones in $\mathbb R^n$ can be non closed but I am not convinced and cannot manage to find a counter-example, can you find any? By ...
1
vote
1answer
9 views

Compactness of stabiliser subgroup of automorphism group of an open convex cone

I have a question about the following proof in Analysis on Symmetric Cones by Faraut and Koranyi (p.5). Let $G(\Omega)$ be the automorphism group of an open convex cone $\Omega$. For any point $a\in \...
0
votes
0answers
16 views

What is the normal cone of the constraints of a quadratically constrained quadratic programming(QCQP)?

$$\begin{array}{ll} \text{minimize} & f_0(x)\\ \text{subject to} & f_i(x) \leq 0\end{array}$$ where $$f_i (x) := (A_ix+b_i)^T(A_ix+b_i)-c_i^Tx-d_i$$ How can I calculate the normal cone of ...
0
votes
0answers
132 views

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 ...
0
votes
1answer
140 views

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 $x^* +td \in A, \forall t \in (0,\delta) \}$ ...
0
votes
0answers
115 views

Is it a simplicial cone always a pointed cone?

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 ...
0
votes
1answer
19 views

$[A^{-1}(\mathbb{R} ^m _{-})]^* = A^{T}(\mathbb{R} ^m _{+})$ 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 $...

1
2 3 4 5