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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12 views

Polyhedral cone face lattice computation using a software. [closed]

I am looking for a software to compute the face lattice of a polyhedral cone other than Macaulay 2 and NORMALIZ.
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What is the theory of the cones of minorant functions?

As a non - mathematician, I face some difficulty in understanding concepts such as convex cone, minorant/majorant function. The question is What is the theory of the cones of minorant functions? And ...
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56 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 ...
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25 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 ...
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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 ...
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6 views

Union of cone of feasible directions

The definition of cone of feasible directions: $D(S,x)=\{ y\in E| \exists \delta>0 \text{ such that } \forall t \in (0,\delta), x+ty\in S\}$. Let $S_1\subset E $, $S_2\subset E$, and $x\in S_1 \...
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9 views

Separation argument and polar cone

Let $S\subset \mathbb{R^n}$, $T\subset \mathbb{R^n}$ be nonempty closed convex cones. Define $C^*=\{ y\in \mathbb{R^n} | y\cdot x \leq 0, \forall x \in C\}$ I am trying to show $(S\cap T)^* \subset ...
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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 \...
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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 ...
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any point in $\mathbb{R}^n$ can be written as a point in the cone $C$ plus a point in its polar cone $C^*$.

Let $C$ be a nonempty convex cone and $C^*$ be a polar cone in $\mathbb{R}^n$. Show that $C+C^*=\mathbb{R}^n$, that is, any point in $\mathbb{R}^n$ can be written as a point in the cone $C$ plus a ...
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105 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 ...
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21 views

the volume between 2 intersecting cones

so imagine that the cones are stage lights. so I want to find the volume of the intersection with the 2 cones. the variables are the x, y, and z angles, and the radius and heights of the cones and the ...
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37 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 $x^* +td \in A, \forall t \in (0,\delta) \}$ ...
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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 ...
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17 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 $...
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linear independence, convex hull, and cone

I am trying to show that for any set, $A\neq \{ 0\}$ and any $x\in co(cone A) \subset \mathbb{R^n}$, there exists linearly independent set $A' \subset A$ such that $x\in co (cone A')$. Here, co(A) ...
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Show that $a^Tz\ge b$ for $z\in K$ if $a^Tz>b$ for $z\in \operatorname{int}(K)$ with $K$ being a convex cone

I'm working on a proof and got stuck with the following situation. Given are two convex cones $K_1, K_2\in\mathbb{R}^n$ with non-empty interior such that $\operatorname{int}(K_1)\cap \operatorname{int}...
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25 views

For full-dimensional cone $K$, $x\in int(K)$, can you take arb. vector $y$ and for some $t$ small enough have that $x-ty\in int(K)$

I'm working on the proof of the following theorem: $K$ full-dimensional, closed, convex cone. $x\in int(K) \iff y^Tx>0 \quad \forall y\in K^*-{0}$ And we're pretty set with the $\Leftarrow$ ...
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Cone of positive Hilbert-Schmidt operators

Let $H,K$ be complex Hilbert spaces and $\operatorname{HS}(H),\operatorname{HS}(K)$ the spaces of Hilbert-Schmidt operators. I will identify $\operatorname{HS}(H)$ with $H\otimes \overline{H}$ (where $...
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1answer
62 views

Given a closed, convex, full-dimensional cone $K$, how do I show that $u\in int(K) \iff u^tx>0 \quad \forall x\in K^*-\{0\}$?

Given a closed, convex, full-dimensional cone $K$, how do I show that $x\in int(K) \iff y^Tx>0 \quad \forall y\in K^*- \{0\} $ ? I've thought about applying the Hahn-Banach separation theorem ...
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135 views

Probability that random variable is inside cone

Suppose $x\in\mathbb{R}^n$ is a random variable with mean $\mu$ and covariance $ \Sigma$. Consider a stochastic convex optimization problem, i.e. an optimization problem with chance constraints, ...
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18 views

is a positive ray is a convex cone?

A definition of a convex cone is given here, could anyone tell me whether a ray $C=\{(x,y): y=mx, x\ge 0,y \ge 0, m=\tan \theta, 0< \theta< 90\}$ on the positive quadrant satisfies the condition ...
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55 views

$x \ge 0, z \ge 0, xz \ge y^2$ for symmetric $2 \times 2$ positive semidefinite matrix

I am told that the positive semidefinite cone in $\mathbf{S}^2$ is $$X = \begin{bmatrix} x & y \\ y & z \end{bmatrix} \in \mathbf{S}^2_+ \iff x \ge 0, \ \ \ z \ge 0, \ \ \ xz \ge y^2,$$ ...
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1answer
57 views

Is the nonnegative orthant a “convex polyhedral cone”?

I am currently studying Convex Optimization by Boyd and Vandenberghe. Chapter 2.2.4 Polyhedra gives the following description of a nonnegative orthant: Example 2.4 The nonnegative orthant is the set ...
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1answer
52 views

Mori cone: extremal ray intersections

On an algebraic surface, much can be said about the Mori cone, or cone of curves. In this question, I will be particularly interested in intersection properties. Several sweeping statements can be ...
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34 views

Show sum of two polyhedral cones is a polyhedral cone?

Polyhedral cone is defined as $C= \{x \in \mathbb{R}^n \mid Ax \geq 0\}$. Let $C_1$ and $C_2$ be two polyhedral cones in $\mathbb{R}^n$. Show that $C_1+C_2$ is also a polyhedral cone. My try: Let $x +...
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1answer
63 views

How is a halfspace an affine convex cone?

Wikipedia says "An affine convex cone is the set resulting from applying an affine transformation to a convex cone" (https://en.wikipedia.org/wiki/Convex_cone) It also says a halfspace is an affine ...
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1answer
64 views

The boundary of the convex hull of squares of skew-symmetric matrices

Let $n \ge 3$, and let $C$ be the convex cone generated by the squares of all real $n \times n$ skew-symmetric matrices. Is $C$ closed in $\mathbb{R}^{n^2}$? What is its boundary? $C$ is a strictly ...
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1answer
36 views

Is a convex cone which is generated by a closed linear cone always closed?

Let $C \subseteq \mathbb{R}^n$ be a closed cone which contains zero. (i.e. $\lambda C \subseteq C$ for every $\lambda \ge 0$). Let $P(C)$ be the convex cone generated by $C$, i.e. the set of all ...
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1answer
44 views

Does every negative semidefinite matrix lie in the convex cone generated by the squares of skew-symmetric matrices?

Let $C$ be the convex cone generated by all the squares of real $n \times n$ skew-symmetric matrices. Does every negative-semidefinite matrix lie in $C$? I know that every square of a skew-...
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1answer
72 views

Rewrite as second order cone constraint

Can someone please explain how to convert the following into a second order cone programming formulation: $\{(x,y,z,w,u): x,y,z,w \geq 0, (xyzw)^{\frac{1}{2}} \geq ||u||_2^2\}$ $\{(x,y,z,w,u): x,y,...
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30 views

Closedness of Cone Generated by Closed Set

Let $ \phi \neq S \subset \mathbb{R}^n$ be a closed set. Let $\bar{x} \in S$. Define the cone $C_{\bar{x}} = \{y \in \mathbb{R}^n :| \exists \lambda \ge 0,\space x \in S \space s.t \space \space y=\...
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61 views

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 ...
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1answer
67 views

Proof dealing with the union of cones and the intersection of polar cones

I need to solve the following problem: Let $S_1$ and $S_2$ be two cones. Let $P(S)$ be the notation for Polar Cone (P) of $S$. Let $S_1, S_2\in \mathbb{R}^n$. (i) Show that $S=S_1 \cup S_2$ is a ...
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1answer
32 views

extreme vector in the positive cone of $C*$-algebra

The definition of extreme vector is defined in the screenshot.Suppose $A$ is a unital $C^*$-algebra,$A^{+}$ is the set of all positive elements in $A$,does there exist an extreme vector in $A^{+}$?
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42 views

Tangent space of complex positive semidefinite cone at $E_{11}$

Take the cone of all Hermitian positive semidefinite matrices. I am interested in the tangent spaces of the boundary. Especially, what is the tangent space at the point $E_{11}$, the matrix with a $1$ ...
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1answer
27 views

Example of Pointed Cone

A Cone $K$ is pointed if $k \in K$ and $-k \in K$ then $K=0$. but in this example how to check this is pointed or not. $ K = \{ (x,y) \in R^2 : \left|x\right|\leq \left|y\right| \} $
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Describe the cone of nonnegative polynomials in nonnegative variables

I was reading an article on the Buffalo way to prove some Olympiad style inequalities. In the examples given, the claims were reduced to seeing that certain polynomials in nonnegative variables take ...
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74 views

Dual cone to exponential cone

Given $$K_{exp} = \left\{ (x, y, z) | y > 0, ye^{\frac x y} \leq z \right\} \cup \left\{ (x, y, z) | x \leq 0, y = 0, z \geq 0 \right\} $$ Find $K^*_{exp}$ $$ K^*_{exp} = \left\{ (a, b, c) | ...
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Normal Cone to a closed convex cone $\mathcal{K}$ at $x \neq 0$, $x \in bdry(\mathcal{K})$

I want to show that the normal cone is the set of all vectors $w$ such that $\langle w, x \rangle = 0 $. Geometrically in 2D for example, it is trivially true. Do I need to show this or is it true ...
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1answer
90 views

Find the dual cone $K^*_{m+}$ of $K_{m+} = \{ x \in \mathbb{R}^n \mid x_1 \ge x_2 \ge \dots \ge x_n \ge 0\}$

We define the monotone nonnegative cone as $$K_{m+} = \{ x \in \mathbb{R}^n \mid x_1 \ge x_2 \ge \dots \ge x_n \ge 0\}$$ i.e. all nonnegative vectors with components sorted in nonincreasing order. ...
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2answers
27 views

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 $\forall x \neq 0 \in K$,...
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17 views

How can we determine the normal cone to $\left\{w\in\mathcal L^2(\mu)\times \mathcal L^2(\mu):w_1+w_2=1\right\}$?

Let $(\Omega,\mathcal A,\mu)$ be a probability space, $$S:=\left\{w\in\mathcal L^2(\mu)\times \mathcal L^2(\mu):w_1+w_2=1\;\mu\text{-almost surely}\right\}$$ and $w\in S$. How can we determine the ...
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1answer
153 views

Closure of a cone

Suppose a set $\mathcal{X}$ is closed and bounded, and define $\mathcal{K}_\mathcal{X} = \{(x,t) : t > 0, \frac{x}{t} \in \mathcal{X} \}$. Show that: $$\bar{\mathcal{K}}_\mathcal{X} = \mathcal{K}_\...
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23 views

Can we represent a “well behaved” cone as a set of squares?

For instance $\mathbb{R}_+^n$ is the set of $(x_1^2,\ldots,x_n^2)$ where $x_i$ are arbitrary real numbers. Similarly the nonnegative semidefinite cone is the set of $X^TX$ where $X$ is an arbitrary ...
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1answer
56 views

Characterising the cone of elements whose inner product with $v \otimes v$ is non-negative

Let $V$ be an $n$-dimensional real inner product space. Consider the space $V \otimes V$, endowed with the tensor product metric, i.e. $$ \langle v_1 \otimes v_2 , w_1 \otimes w_2\rangle := \langle ...
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2answers
201 views

Volume of a frustum given the bottom radius and the top cone height.

A cone with base radius 12 cm is sliced parallel to its base, as shown, to remove a smaller cone of height 15 cm. If the height of the smaller cone is three-fourths that of the original cone, what ...
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1answer
101 views

Determining if a Vector is a member of a Convex Hull

Edit: Here is how the following sets are created: $S$ is the set of all $n$-dimensional, multilinear trinomials that are strictly greater than $0$ on the interval $[0,1]^n$. $T$ are all miltilinear ...
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1answer
87 views

Proof that the image of a closed cone under a linear transformation is a closed cone [duplicate]

The question is: Let $T : \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear map. Let $C$ be a closed subset of $\mathbb{R}^n$ which is also a cone. Prove that if $\ker(T) \cap C = \{0\}$ then $T(C)$ ...
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1answer
60 views

If $\epsilon \in (0,1]$ and $\tan^2\theta \leq \epsilon$, then what are the admissible values of $\theta$ in terms of $\epsilon$?

Let $\epsilon \in (0,1]$, and $\theta$ is such that $\tan^2\theta \leq \epsilon$. What are the admissible values of $\theta$ in terms of $\epsilon$? The motivation for the question is the following: ...