Mathematics Stack Exchange
Mathematics Stack Exchange

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

286 questions
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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 "simple" explicit formula for computing the…
M. Winter
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Definition of a convex cone

In the definition of a convex cone, given that $x,y$ belong to the convex cone $C$,then $\theta_1x+\theta_2y$ must also belong to $C$, where $\theta_1,\theta_2 > 0$. What I don't understand is why there isn't the additional constraint that…
Undertherainbow
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Find a nonnegative basis of a matrix nullspace / kernel

I have a matrix $S$ and need to find a set of basis vectors $\{\mathbf{x_i}\}$ such that $S\mathbf{x_i}=0$ and $\mathbf{x_i} \ge \mathbf{0}$ (component-wise, i.e. $x_i^k \ge 0$). This problem comes up in networks of chemical reactions where $S$ is…
Name
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The space of positive semidefinite $n \times n$ matrices is a cone

The positive semidefinite cone is defined as follows $$\mathbb{S}^{n}_{+} := \left\{ \mathbf{X}\in\mathbb{S}^{n}: \mathbf{X}\succeq\mathbf{0} \right\}.$$ To my knowledge, this definition imposes two restrictions on ${\bf X}$: $\mathbf{X}$ is a…
pej
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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 $\{\omega_s:s\in S\}$ be the dual basis of $\{\alpha_s:s\in…
Brauer Suzuki
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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, meaning there is a small, but finite probability,…
Josh Pilipovsky
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Characterizing duals of cones that are linear images of the positive semidefinite cone

Let $M_n$ denote the space of $n\times n$ matrices over complex numbers. The space of self-adjoint matrices is denoted $$ M_n^{sa} = \{A\in M_n\, :\, A^*=A \}, $$ where $A^*$ denotes the conjugate transpose of $A$, and the cone of positive…
Luftbahnfahrer
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Determining whether or not a vector is in a cone

Determine whether or not the vector $\langle 0,7,3 \rangle$ belongs to the cone generated by $$\langle 1,1,1\rangle \qquad \langle -1,2,1\rangle \qquad \langle 0,-1,1\rangle \qquad \langle 0,1,0\rangle$$ That is, I am asked to determine whether…
user407200
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Why can't a convex cone have more than one extreme point?

Let $S$ be a set. I define an extreme point as a point $x \in S$ that cannot be defined as a convex combination of two distinct points $x_1, x_2 \in S$, i.e., if $x=\lambda x_1+(1-\lambda) x_2$ for $x_1 \neq x_2$, and $\lambda\in [0,1]$, then…
Vinith
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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 translate to an equivalent conic optimization…
IOS_DEV
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Pappus centroid theorem and Hypercones.

The volume of a straight cone in $\mathbb R^3$ is usually find adding the circular sections orthogonal to the height. If the base has radius $R$ and the height is $h$ we have: $$ V_{C3}=\int_0^h \pi r^2 dz=\pi \int_0^h \frac{R^2}{h^2}z^2 dz=\pi…
Emilio Novati
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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 interesting examples of $K$ and $K^*$, especially when…
max_zorn
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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 in terms of a suitable norm.
noirritchandra
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Slant cone volume problem

I was given a a problem to solve, I thought I solved it but my answers don't look like the ones provided. The Problem A cone with radius of base r and height h, is stretched to the left and right by length a and b respectively such that height of…
Nerdrigo
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What Is a Pointed Cone Intuitively? How Could One Visualize It?

A cone $K$, where $K ⊆\Bbb R^n$ , is pointed; which means that it contains no line (or equivalently, $(x ∈ K~\land~ −x∈K) ~\to~ x=\vec 0$.
MORAMREDDY RAKESH REDDY
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