# Questions tagged [cones]

The cones tag has no usage guidance.

50
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### 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 ...

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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 ...

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**1**answer

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 ...

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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, $\...

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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 ...

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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 ...

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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 ...

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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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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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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 ${\...

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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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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 $\{\...

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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 ...

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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 ...

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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)=\...

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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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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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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\|...

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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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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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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 $\gamma > 0$ ...

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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 ...

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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 ...

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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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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 ...

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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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### 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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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, ...

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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 $\...

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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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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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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
$C \cap (-C) = \{ 0 \}$.
...

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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 ...

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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?
...

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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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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 ...

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**1**answer

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 ...

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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 ...

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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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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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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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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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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
$V_+ \cap (-V_+) = \{0\}$...

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**1**answer

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 ...

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**1**answer

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 $\mathbf{R}^{n}-0$....

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**3**answers

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 $...