Questions tagged [cones]

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28 views

Limiting normal cone of a n-sphere-reduced n-cube

Imagine you have a n-cube $C = \{w \colon \|w\|_\infty \leq 1\}$. Since $C$ is convex, it is simple to compute the normal cone to $C$. Imagine now you have the following set $S = \{w \colon \|w\| \leq ...
4
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1answer
119 views

Finding Motzkin's original paper on copositive quadratic forms

I am currently in the process of writing my thesis about copositive matrices and would like to write a chronological narrative about the ascent of these matrices to the prominent place they have today ...
1
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1answer
75 views

Intersection of a closed convex cone with the non-negative orthant

Suppose I have a closed convex cone $C\subseteq \mathbb R^n$ and suppose that for every $x$ in the non-negative orthant $\mathbb R_{0+}^n$ there is a $y\in C$ such that $x\cdot y>0$ (with the ...
9
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1answer
234 views

Closedness of linear image of positive L1 functions

Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\...
3
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0answers
61 views

How to find the dimension of the polar cone of a convex cone generated by some given vectors

Suppose we have access to a generating set $\{v_1, ..., v_k\}\subseteq\mathbb{R}^n$ of the convex cone $C=cone(v_1, ..., v_k)$, where $cone(\cdot)$ is the conical hull (i.e. nonnegative span) of ...
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29 views

How to project a sequence on the univariate moment cone

Folowing closely Schmüdgen, K. (2017). The moment problem (Vol. 9). Berlin/New York: Springer., Chapter 10, Section 2, for a bounded interval $[a,b] \in\mathbb R$ and $m\in\mathbb N$, define the ...
4
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0answers
65 views

Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism

Let $R$ be any ring and let $A\to B\to C\to [1]$ and $A'\to B'\to C'\to [1]$ be distinguished triangles of complexes of $R$-modules. Let $f:A\to A'$, $g:B\to B'$ and $h:C\to C'$ be morphism of ...
2
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1answer
78 views

When does the map from a normed vector cone to its double dual preserve norms?

If $V$ is a normed vector space then the natural map from $V$ to its double dual $V''$ is norm-preserving as follows from Hahn-Banach theorem. This is well-known. Now assume that P is just a vector ...
5
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1answer
145 views

A characterisation of faces of rational polyhedral cones

This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could ...
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118 views

Gaussian mean width of normal random cones

Suppose $1 \leq n < m < \infty$ are integers. For $g \sim \mathcal N(0, I_n)$ define the gaussian mean width of a non-empty set $T \subseteq \mathbb R^n$ by $$ w(T) := \mathbb E \sup_{x \in T} \...
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1answer
264 views

Why does every chain complex have a map into its cone?

In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...
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1answer
396 views

Affine cone example

Consider the complete intersection ideal ${\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\...
2
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1answer
86 views

Do highly symmetric cones have “small” supporting hyperplanes?

Let $C$ be a full-dimensional cone in $\mathbb{R}^{d}$, defined as the positive span of $c = {n \choose 3} \gg d$ vectors. $C$ is highly symmetric in the following sense: each such vector is labelled ...
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62 views

Formula for exponential integral over a cone

While reading 'Computing the Volume, Counting Integral points, and Exponential Sums' by A. Barvinok (1993), I came across the following: "Moreover, let $K$ be the conic hull of linearly independent ...
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1answer
265 views

Closed convex cone - equivalence of definition via closure and via infinite sums

I have a set $P$ of points in a Banach space. Consider the following two cones: The closure of the set of all (finite) nonnegative linear combinations of $P$. (I.e., the topological closure of $\{\...
8
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2answers
932 views

How many cones with angle theta can I pack into the unit sphere?

Given a unit sphere (radius 1), I would like to know how many cones I can pack into this unit sphere. Restrictions: The top of the cone needs to be in the center of origin. The bottom of the cone ...
4
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109 views

Name for facet of a cone containing all but one edge

Let $C \subseteq \mathbb R^n$ be a polyhedral cone, so generated by its edges ($1$-dimensional faces) and $F \subseteq C$ a facet (codimension $1$ face) of it containing every edge except $e$. In ...
2
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1answer
100 views

Separation of two pointed polyhedral cones using hyperplanes generated by facets

Let $C_1$ and $C_2$ two pointed (that is, with vertex in $0$) polyhedral cones in $\mathbb{R}^n$ with $\dim(C_1)=\dim(C_2)=n$. If $$\mbox{relative interior}(C_1)\cap \mbox{relative interior}(C_2)=\...
3
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1answer
76 views

Faces of polyhedral cones and open immersions of affine toric schemes

Let $V$ be an $\mathbb{R}$-vector space of finite dimension, let $N$ be a $\mathbb{Z}$-structure on $V$, and let $M$ be its dual $\mathbb{Z}$-structure on the dual space $V^*$. Let $\sigma\subseteq V$...
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206 views

Complexity of conic optimization problems

I am interested in bounding the computational complexity of the interior points method for solving a generic conic problem of the form \begin{equation} \min_x \left\{ c^T x : \mathcal{A}x-B\in\mathbf{...
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1answer
785 views

1st Order Nonlinear PDE: Understanding Envelopes and Monge Cones

I have a question about envelopes of surfaces. In a book I am reading the following: Suppose $S_a$ is a one parameter family of surfaces in $R^3$ given by $z=w(x,y;a)$ where $w$ depends smoothly on ...
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1answer
618 views

Projection onto the second-order cone [closed]

I'm having difficulties in proving that the projection of $$(s,y)\in R \times R^{n}$$ onto the second-order cone $$Q^{n+1} = \{(t,x) \in R \times R^n : \|x\|_2 \leq t \}$$ is $$ \frac{s+\|y\|_2}{2\|y\|...
3
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1answer
79 views

Cone construction for Birkhoff Hopf theorem

Let $M$ be a matrix such that $\forall i,j$ $M_{ij}\geq 0$ and suppose that $M$ is irreductible. 1 - Is there a natural change of basis such that the new matrix became strictly positive : $\forall i,...
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0answers
65 views

Are these convex cones polyhedral?

I'm actually playing with some convex cones, and I would like to know if there is a chance they would be described by a finite number of inequalities. Let me introduce some notation first. Let $n\...
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1answer
69 views

Showing that $Df_x H_x^\gamma \subset H_{f(x)}^{\lambda \mu^{-1} \gamma}$, where $H_x^\gamma$ is a family of horizontal cones

Let $M$ be a smooth manifold, $U \subset M$ an open set, $f : U \to M$ a $C^1$ diffeomorphism onto its image and $\Lambda \in U$ a hyperbolic set for $f$. Fix a sufficiently small $\gamma > 0$ ...
1
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1answer
166 views

nonnegative solution of nonhomogeneous under-determined linear system of equations

For a set of under-determined linear equations, I was wondering if there is any closed form for all non-negative solutions? Is there a way to analytically characterize the feasibility set of such ...
3
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0answers
220 views

Are “vector spaces” over a smooth scheme with constant fiber dimension locally free?

I've got the follow question which drives me almost crazy as the answer seems to be simple. Given a morphism $p:V\to S$ of schemes of finite type over some base field. Assume that $p$ has all the ...
4
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2answers
426 views

Maximal cones and lexicographic orderings

Let $V$ be a real vector space. It is well known that, given a totally ordered basis of $V$ (say $(b_i)_{i\in I}$ where $I,<$ is totally ordered), $V$ is totally ordered by the lexicographic ...
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0answers
109 views

Finding generators of symmetric cones

I have a bunch of vectors $\mathbf v_i$ in $\mathbb R^n$. I would like to consider the cone $C$ spanned by these vectors, together with all the other vectors that can be obtained by permuting the ...
5
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2answers
441 views

Positive Elements of a $\ast$-Algebra

In a $C^*$-algebra ${\cal A}$, a positive element is a one of the form $aa^*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a cone, and that for $a,b$ two non-zero ...
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0answers
69 views

Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?

Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$, $$f(v) = A[v]_+ + B[-v]_+$$ surjective? Here $[.]_+$ is an elementwise ...
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0answers
62 views

How does the singular surfaces obtained when the border of a Euclidean set becomes a point look like?

I'm curious to understand in several manner, what is the metric geometry of the metric space homeomorphic to a sphere, obtained from a compact convex set $K\subset R^2$ with the Euclidean distance, ...
3
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1answer
371 views

The sign of the mean curvature on convex cones in three dimensions

My question is as follows: It is known that a closed smooth curve in $\mathbb{R}^2$ is convex iff its (signed) curvature has a constant sign. I wonder if one can characterize smooth convex cones in $\...
3
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1answer
149 views

Dual cone of 'positive' Bochner integrable functions

If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}_+)$. It is known that the ...
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0answers
79 views

A version of isotone projection cones

We write $a \succeq b$, where both $a, b \in \mathbb{R}^n$, as a shorthand for $a_i \ge b_i$ for all $1 \le i \le n$. Let $C$ be a closed convex cone in the first orthant of $\mathbb{R}^n$ and denote ...
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2answers
1k views

Base of a cone in a vector space: can one always choose a convex base?

Let $C$ be a pointed convex cone in a vector space $V$. This means that $C$ satisfies the three following axioms: $C + C \subset C$, $\mathbb{R}_+ \cdot C \subset C$, and $C \cap (-C) = \{ 0 \}$. ...
5
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2answers
609 views

Decomposing polyhedral cones into “direct sums” and a polynomial

This question consists of two parts. I'm not breaking it up into two separate ones because posing the second question would essentially require me two rewrite the first one. Also, to some extent, the ...
3
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2answers
792 views

How to (efficiently) find intersection of two polyhedral cones?

I have two polyhedral cones represented by their rays. I am looking to find their intersection, which would also be a polyhedral cone, hopefully efficiently. Does anybody know a way to do that? ...
3
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3answers
259 views

Sensitivity analysis in conic optimization

I have a conic optimization of the form: $$\min_x \langle c, x \rangle,\ \text{s.t.}\ Ax = b,\ x \in K.$$ where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a ...
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1answer
474 views

Is dual cone unique? [closed]

Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones, $A^\ast=C,$ $B^\ast=C,$ can we state that $A=B$? Is the dual cone of a cone is unique? the definition ...
2
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1answer
2k views

Kleiman's and Nakai-Moishezon's ampleness criteria

I would like to work out a simple example to understand the relation between Kleiman ampleness criterion and Nakai-Moishezon ampleness criterion. Namely, let $X$ be the blow-up of $\mathbb{P}^{2}$ at ...
2
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2answers
193 views

Representation of Banach spaces partially ordered by solid, normal, minihedral cones

I've been using the representation result below, from Krasnosel'skij/Lifshits/Sobolev; Positive Linear Systems---The Method of Positive Linear Operators. Heldermann Verlag, 1989. Theorem. Let $E$ be ...
2
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0answers
102 views

Suprema and infima in spaces ordered by non-normal cones

Background We shall call a subset $V_+ \subseteq V$ of a Banach space $V$ a cone if $V_+$ is closed, $\alpha V_+ \subseteq V_+$ for all $\alpha \geqslant 0$, and $V_+ \cap (-V_+) = \{0\}$. Cones ...
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0answers
232 views

Linearization of cones

Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?
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1answer
969 views

When is a matrix similar to a non-negative matrix?

Consider a real square matrix $A$ of size $n\times n$. Under which conditions on $A$ does there exist a row-stochastic matrix $U$ (non-negative, rowsums = 1), such that $A'=U^{-1}AU$ is a non-negative ...
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0answers
72 views

Which matrix/operator in a cone has the largest negative spectral part?

Background: Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+}-A_{-}$, where $A_{+}$ ...
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0answers
372 views

Is it true that a solid, minihedral cone in infinite dimensions cannot be regular?

Background Consider a real Banach$^1$ space $V$. We'll call a subset $V_+ \subseteq V$ a cone if $V$ is closed, $\alpha V_+ \subseteq V_+$ for every $\alpha \geqslant 0$ and $V_+ \cap (-V_+) = \{0\}$...
2
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1answer
528 views

Covering the cone of positive semidefinite matrices by intervals

Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices? How about a general convex cone? For the finite case the ...
3
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1answer
492 views

When are cones of matrices “generated” by vectors?

The cone $P_{n}$ of positive semidefinite matrices of order $n$ can be represented in this form: $P_{n}=\{A|\forall x\geq 0: \langle A,xx^{T}>0 \rangle \}$ with $x$ running over $\mathbf{R}^{n}-0$....
5
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3answers
1k views

What fraction of a sphere's volume lies within a cone?

Let $B \subset \mathbb{R}^n$ be a collection of $n$ (not necessarily independent) unit vectors which we will label $v_1,\ldots, v_n$ for convenience. The cone $K_B \subset \mathbb{R}^n$ associated to $...