# Questions tagged [cones]

The cones tag has no usage guidance.

63
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### Order bounded version of monotone complete $C^*$-algebras

Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...

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2
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### Subdifferential of a convex function admits a continuous selection

Let $F$ be a continuous convex function on $\mathbb{R}^n$.
If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is ...

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### 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 ...

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### Existence of a smooth extension

In the three dimensional Euclidean space $\mathbb R^3$ let us define the hypersurface
$$ S= \{(x,y,z) \in \mathbb R^3\,:\, z^2= x^2+y^2\}.$$
Suppose that $f \in C^{\infty}(S)$. Does there exist $u\in ...

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### Conditions for including cones

Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...

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### Are the automorphism groups of simple symmetric cones algebraic groups?

This question arises when I tried to understand Chapter 2 of the celebrated book "Smooth compactification of locally symmetric varieties" by Ash–Mumford–Rapoport–Tai.
The setting is as ...

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1
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188
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### Semisimple Lie algebra and convexity

There is any relation between semisimple lie algebras and symmetric cones? I'm saying this because the classification of the euclidean Jordan algebras, dual by Koecher-Vinberg theorem to homogeneous ...

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### Polynomial-time algorithm for exact projection to polyhedral cone

Given $c \in \mathbb{R}^d$ and $A \in \mathbb{R}^{n \times d}$, project $c$ to the polyhedral cone $\{x \in \mathbb{R}^d \mid A x \leq 0\}$. Is there an algorithm that outputs an exact solution to ...

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### Non-proper orthant automorphisms

Given a real vector space $V$ of dimension $d$, it is known that the automorphism Lie group of the nonnegative orthant $\mathcal{O}^+_d$ can be described just as $$\mathrm{Aut}(\mathcal{O}^+_d)=\...

3
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1
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### Image of a quadratic form is a closed cone

Let $Q : E \to F$ be a quadratic form induced by a symmetric bilinear form $B : E \times E \to F$ defined in a finite dimensional real normed vector space $E$, with values in the normed vector space $...

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### cone structure of complement of hyperplanes

I want to know if in $\mathbb{R}^{m+3}$ we consider the following hyperplanes:
\begin{cases}
(1-g)y-\sum_{i\in I}x_i=0, & \text{if $I\subset\{1,\cdots,m+2\}$},|I|=g\\
gy-\sum_{i\in I}x_i+\...

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1
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### Interpolation of normed spaces *vs* geometrical mean of positive matrices

Two $n\times n$ positive definite symetric matrices $A,B$ define two normed spaces $E_A=({\mathbb R}^n;\|\cdot\|_A)$ and $E_B=({\mathbb R}^n;\|\cdot\|_B)$, where
$$\|x\|_A=\sqrt{x^TAx},\qquad \|x\|_B=\...

0
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148
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### Is $(K^*)^{**}=(K^{**})^*$ for any cone $K$?

I'm considering the dual cone $K^*$ of a non-convex cone $K$. I came up with a theory that $K^{**}$ is the closure of convex hull of $K$. Then I wonder whether $(K^*)^{**}=(K^{**})^*$ holds for any ...

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### What is the convex cone generated by the pair of rank 1 matrix and its eigenvector?

I'd like to know what is the convex cone generated by $\left\{ (h h^T, h) : h \in \Bbb R^{d\times1} \right\}$. It is known that $$\mathrm{cone} \left\{h h^T : h \in \Bbb R^{d \times1} \right\} = S_+^d$...

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### Can min-max be set up around a minimal cone?

Let me state my question in very loose terms to start, then give some details and restate it in more precise terms at the bottom.
Question. Given a regular minimal cone $\mathbf{C}$, can one set up a ...

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151
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### 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
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### 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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### 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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### 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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### 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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### 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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### 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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### 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} \...

6
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1
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### 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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### 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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1
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### 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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### 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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1
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343
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### 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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2
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### 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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### 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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### 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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### 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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### 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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1
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### 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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1
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### 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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### 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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### 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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1
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### 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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1
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...