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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.

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Tangent cone to a convex set is a convex cone

Let $Y \subseteq \mathbb{R}^n$ be a convex set and let $\bar{y} \in Y$. The tangent cone to $Y$ at $\bar{y}$, denoted $T(Y, \bar{y})$, is the set of all limits of the form $h = \lim t_{l}(y_{l} - \bar{...
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Describe all cones in $\mathbb R^n$ that are simplicial and cosimplicial.

For any finite set of vectors $(v_1, \ldots, v_k \in \mathbb{R}^n)$, let's define: $Cone(v_1,…,v_k)= \{λ_1v_1+…+λ_kv_k,∣,λ_1,…,λ_k \ge 0\}$ and $Poly(v_1,…,v_k)= \{ x \in \mathbb{R}^n ∣(v_i, x) \ge 0, ...
PikaPika's user avatar
3 votes
2 answers
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The normal cone to a ball at a boundary point of the ball

I got stuck in a problem that in the end I needed to know what is the normal cone to a ball $B = \left\{ x \in \mathbb{R}^n : \lVert x \rVert \leq 1 \right\}$ (here $\lVert \cdot \rVert$ is an ...
Raul Fernandes Horta's user avatar
1 vote
1 answer
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Prove if the polyhedron $A x \leq b$ is bounded, then the cone hull of all rows of $A$ is $\mathbb{R}^d$

Let $ P = \{x \in \mathbb{R}^d:Ax \leq b\}$ be a polyhedron, where $A = \begin{bmatrix} a_1^T\\a_2^T\\...\end{bmatrix}$, then the cone hull of $(a_1,a_2,\cdots)$ is $\mathbb{R}^d$. How to prove this ...
Nekomiya Kasane's user avatar
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Basis of the intersection of a cone and its dual.

Let $a_1,\dots, a_m$ be vectors in $\mathbb R^n$, let $A\in\mathbb R^{n\times m}$ whose columns are $a_i$s. The cone generated by $A$ (denoted $\operatorname{cone}A$) is $C=\{ Aw: 0\leq w \}$, its ...
P. Quinton's user avatar
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Questions related to Cones and Subspaces of Euclidean Space

Cone: A subset $ S \subseteq \mathbb{R}^n$ is a cone if $\alpha \geq 0 \implies \alpha S \subseteq S.$ Polar: A Polar $K^*$ of a cone $K$ is a closed convex cone such that $$K^*=\{y \in \mathbb{R}^n \...
Mani's user avatar
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If a convex cone K and a linear subspace L are such that $K\cap L=\{0\}$, is it true that the intersection of their polars has a non-zero vector?

A bit more details: I'm trying to prove that if the first condition holds, then the following statement is also true: $$\exists z\neq 0 \text{ such that }z\in K°\cap L^\perp.$$ That second condition ...
Pablo's user avatar
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Can this constraint be cast as a second order cone constraint?

Can someone please explain if it possible to convert the following constraint into a second order cone programming formulation: $xy \ge ay + b$ Here $x,y$ are non negative decision variables, $a,b$ ...
Tuong Nguyen Minh's user avatar
1 vote
1 answer
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A statement regarding tangent cones and polyhedricity and variational inequalities (need help with a proof)

I'm reading this thesis on page 182. We work in a Hilbert space $H$, and $y \in H$ solves the variational inequality: $$y \in K : \langle Ay-f, v-y \rangle \geq 0 \; \forall v \in K$$ for some given ...
StopUsingFacebook's user avatar
1 vote
1 answer
116 views

Polar cone of a closed convex cone in $R^4$ defined by a convex inequality constraint

Let $c>1$ be a constant. Consider points in four dimension with coordinates $(x,y,z,p)\in R_{\ge 0} \times R_{\ge 0} \times R \times R_{\ge 0}$ and the cone $$K = \{ (x,y,z,p)\in R_{\ge 0} \times ...
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Closedness of a cone

Let $N\in\mathbb{N}$ and $P$ be a symmetric pattern; i.e. $P$ is a subset of $\{1,\dots,N\}\times\{1,\dots,N\}$ such that $(i,i)\in P$, for all $i\in\{1,\dots,N\}$, and $(i,j)\in P$ if and only if $(j,...
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unit vector in recession cone

The recession cone of $\Omega$. If $d_u$ is a unit vector, show that $d_u\in R(\Omega)$ iff there exists unbounded sequence $\{ w^k \}_{k=1}^\infty\subset \Omega$ satisfying $$\frac{w^k}{\|w^k\|}\to ...
Eve_11037's user avatar
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How to design a Riemannian metric for the interior of a proper cone that leaves it a Hadamard manifold?

Let $V$ be a finite-dimensional vector space. Let $K \subset V$ be a proper cone. That is, $\alpha K \subset K$ for $\alpha \geq 0$ (cone) $K$ is closed $K$ is convex $K \cap (-K) = \emptyset$ (...
Spencer Kraisler's user avatar
1 vote
2 answers
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GM and AM convex cone

Let $\displaystyle K_\alpha = \{x \in \mathbb{R}^n| \sqrt[n]{x_1x_2...x_n} \ge \alpha\frac{x_1 + x_2 + ... + x_n}{n}, x_i \ge 0\}$ and $\alpha \in [0, 1]$. Show that $K_\alpha$ is a convex cone. I ...
GeoArt's user avatar
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Intuition for the Cone Lemma

The Cone Lemma. If a system of homogenous linear equations with integer coefficients has a positive real solution, then it also has a positive integer solution. This is proved in Proofs from THE BOOK,...
Joseph O'Rourke's user avatar
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1 answer
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Tangent cone of boundary points of polyhedron

I am reading the paper Positive Invariance Condition for Continuous Dynamical Systems Based on Nagumo Theorem, and specifically I am concerning Theorem 3.1, where the tangent cone at boundary points ...
happyle's user avatar
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Does the vector space of differences of quantile functions have a neat characterization?

Consider the convex cone of quantile functions of random variables on the real line (with finite second moment), that is $$ C := \{ Q_{\mu}: \mu \in \mathcal P_{(2)}(\mathbb R) \}, $$ where $Q_{\mu}(p)...
ViktorStein's user avatar
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the case of separating N disjoint convex cones [closed]

I am learing about the separation theorem : separation between a (closed) convex set $X$ in $\mathbb R^n$ and a vector outside $X$. I know about it but what are some examples where separation is used ...
analysis lover's user avatar
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1 answer
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How could I check to see if a point is inside a cone projected by another point?

I use CesiumJS, a javascript library, to render entities in a 3-dimensional space. I need to detect if one entity is "viewing" another entity. My initial thought is to use a cone to ...
AndyPet74's user avatar
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2 answers
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A question about ordered vector spaces

Relevant Definitions: Definition: Let $X$ be a set. A partial order $\leq$ on $X$ is a relation that is reflexive, anti-symmetric, and transitive. Definition: Let $\leq$ be partial order on a real ...
Mud's user avatar
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basis of monoid of integral vectors

Suppose that $M\in\mathbb{Z}^{n\times k}$ is a matrix of rank $k<n$. How can I obtain a set of vectors $b_1,\ldots,b_k\in\mathbb{Z}^k$ (if exists) such that each row of $M$ is a non-negative ...
Brauer Suzuki's user avatar
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Adjacent extreme rays of a polyhedral cone

I am looking for an algorithm that could determine adjacent extreme rays of a polyhedral cone (extreme rays that share a face), given the set of inequalities defining the cone and an extreme ray for ...
user881461's user avatar
1 vote
0 answers
59 views

The number of faces of a pointed convex polyhedral cone

Let C be an n-dimensional pointed convex polyhedral cone with a uniqe frame {a1.a2.......,ar}, where ais are extrem half-lines. Is there a formula for the number of r-faces in C? Let me state the ...
Masahiro Fujimoto's user avatar
1 vote
0 answers
50 views

Symmetric cones and symmetric spaces

I start by stating what I think I understood on symmetric cones (https://en.wikipedia.org/wiki/Symmetric_cone). Let $\mathcal{C}$ be a symmetric cone in a vector space $V$. There are Riemannian ...
Chevallier's user avatar
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Equivalence of polar cone and polar set

The Wikipedia page on polar cones contains the following statement: "For a closed convex cone C in X, the polar cone is equivalent to the polar set for C". I'm not sure why this is true, ...
Peter's user avatar
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Bouligand and Clarke Tangent cones

Reffering to this question Clarke's tangent cone, Bouligand's tangent cone, and set regularity I'm asking myself if may exist a closed bounded set $S\in\mathbb{R}^2$ and a point $x\in\partial S$ such ...
Mathland's user avatar
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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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1 answer
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Does every pointed polyhedral cone in $\mathbb{R}^n$ generated by its $n+1$ extreme rays contain two extreme ray 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 (...
Veljko Toljic's user avatar
1 vote
1 answer
30 views

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 ...
Chloe's user avatar
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1 answer
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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 ...
vidyarthi's user avatar
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1 answer
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What is the isotropic cone of det in $M_2(K)$?

Consider the application $\text{det}: M_2(K) \to K$ What is the isotropic cone of det in $M_2(K)$? Consider the vector subspace $D$ of the diagonal matrices of $M_2(K)$. Is it true that $M_2(K)=D \...
vitalmath's user avatar
  • 285
0 votes
1 answer
158 views

What does the term 'convex' mean, when we say 'convex cone'.

According to Wikipedia: A convex cone is a a subset of a vector space that is closed under linear combinations with positive coefficients. I wonder if the term 'convex' has a special meaning or ...
Adel's user avatar
  • 103
1 vote
1 answer
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Weyl-Minkowski V-H representation of a cone, with Kronecker products

Conjecture. If the $V$ cone generated by the points in the matrix $A$ has an $H$ representation captured by matrix $B$, and $A = A_1 \otimes A_2$, $B = B_1 \otimes B_2$, where $B_1$ is the $H$ ...
user157623's user avatar
3 votes
1 answer
141 views

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: ...
A. R.'s user avatar
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5 votes
2 answers
215 views

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 ...
IOS_DEV's user avatar
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2 votes
1 answer
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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 ...
MathNewbie's user avatar
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-1 votes
1 answer
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Asymptotic Cone of Product Sets [closed]

We denote the asymptotic cone of a set $S$ by $A(S)$: given $S\subset \mathbb{R}^n$ and a natural number $k$, let $\Gamma(S^k)$ be the smallest closed cone at $0$ containing $S^k=\{x\in S: ||x||\ge k\}...
wxydx00's user avatar
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0 answers
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Subset relation between two polyhedral cones

Suppose we are given two vectors $b_1, b_2 \in \mathbb{R}^n$ and $m \leq n$ vectors $a_1, \dotsc, a_m \in\mathbb{R}^n$ with $ a_1, \dotsc, a_m \notin \text{cone}\, (b_1,b_2) $ $ \text{cone}(a_1, \...
Mar Remi's user avatar
3 votes
0 answers
103 views

Intuitively, what is the matrix core, and how has it proven useful?

I have begun reading some of the (few) references I can find on the concept of a "core" of a non-negative matrix, e.g. here. In brief, my question is as stated in the title. I would like to ...
Jacob A's user avatar
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why the inner product with an element that does not belong to a convex cone is negative

Let us consider a set with an infinite number of vectors $\{v_k \mid k \in \mathbb{N} \}$ with $n$ cordinates, and if we consider the conned convex set containing all theses vectors, denoted by $K$. ...
hanava331's user avatar
2 votes
0 answers
41 views

Is the asymptotic cone $\{0\}$?

I am studying convex analysis at home for self-learning. I focus especially on the structure of closed convex sets. I came across a weird definition of an asymptotic cone in a French (Sorbonne) ...
CHOSM's user avatar
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0 answers
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Is the closure of a positive convex cone positive?

I am studying convex analysis especially the structure of closed convex sets. I need a clarification on something that sounds quite easy but I can't put my fingers on it. Let $E$ be a normed VS of a ...
CHOSM's user avatar
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1 vote
0 answers
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What is an Augmented Vector Space?

I am studying a little bit of Convex Analysis, especially the structure of the closed convex sets (in French). I came across a term that kind of confused me which is "espace vectoriel augmenté&...
CHOSM's user avatar
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1 vote
0 answers
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What is the correct representation of the generalized gamma function?

The NIST Digital Library of Mathematical Functions defines the multivariate gamma function as $$ \Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr% }\left(-\mathbf{X}\...
stollenm's user avatar
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1 vote
0 answers
357 views

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. ...
Addem's user avatar
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0 votes
1 answer
52 views

closed, convex cone C $\in \mathbb{R}^n$ whose linear hull is the entire $\mathbb{R}^2$

I am trying to construct the following example: let $A \in \mathbb{R}^{2x2}$ be symmetric matrix and C $\in \mathbb{R^n}$ a closed, convex cone whose linear hull is the entire $\mathbb{R}^2$ and $x^...
Linchen's user avatar
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1 vote
1 answer
119 views

Does the closed convex hull of a compact set in the interior of a convex cone is still contained in the interior of the cone?

Let $C$ be a convex cone in a Banach space $X$ with nonempty interior. The set $A\subset {\rm Int}C$ is a compact subset, where ${\rm Int}C$ means the interior of $C$. Denote the closed convex hull of ...
Jinxiang Yao's user avatar
0 votes
1 answer
471 views

Is conic programs with exponential cone solvable in polynomial time?

I'm trying to find out some theoretical guarantees of time complexity for my problems. My problem is to minimise a log-sum-exp function. I found that the minimisation of log-sum-exp function can be ...
Jinrae Kim's user avatar
0 votes
1 answer
40 views

Describe the cone generated by vectors $(1,0,0,1), (0,2,3,4), (0,0,3,1), (1,3,2,4), (2,4,6,4)$ by its irredundant linear inequalities.

I'm really not sure on how to tackle this. I have tried considering all matrices comprised of three of the five vectors -- as rows -- and row reducing them, but that hasn't really lead anywhere (so ...
m. lekk's user avatar
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0 answers
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Dual of a norm Cone.

Problem [Example 2.25 Taken from Convex Optimization By Stephen Boyd, Lieven Vandenberghe] In this example, it proves that the dual of a norm cone is the cone of ...
John Smith's user avatar

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