Questions tagged [cones]

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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 ...
Min Wu's user avatar
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4 votes
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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 ...
Stabilo's user avatar
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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 ...
Allen Knutson's user avatar
3 votes
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303 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 ...
Sven Meinhardt's user avatar
3 votes
0 answers
393 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 $V_+ \cap (-V_+) = \{0\}$...
orlandoweber's user avatar
2 votes
0 answers
67 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\...
GreginGre's user avatar
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2 votes
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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 ...
orlandoweber's user avatar
2 votes
1 answer
117 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 ...
Sergey Slavnov's user avatar
1 vote
0 answers
54 views

Determine the class of a non-isomorphic projection of a rational normal scroll as a divisor in a higher dimensional scroll

This is a generalized problem of Theorem 1.1 of Park's and Theorem 1.4 of Nagel's. Consider the vector bundle $E=\mathcal{O}(1)\oplus\mathcal{O}(1)\oplus\mathcal{O}(1)\oplus\mathcal{O}(1)$ on $\mathbb{...
Li Li's user avatar
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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 ...
Chevallier's user avatar
1 vote
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103 views

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 ...
Ali's user avatar
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1 vote
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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+\...
tota's user avatar
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150 views

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 ...
Leo Moos's user avatar
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1 vote
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216 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} \...
bashfuloctopus's user avatar
1 vote
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88 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 ...
Erik's user avatar
  • 81
1 vote
0 answers
342 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{...
mermeladeK's user avatar
1 vote
1 answer
199 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 ...
user avatar
1 vote
0 answers
68 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, ...
Nicolas Juillet's user avatar
1 vote
0 answers
80 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 ...
John Wong's user avatar
  • 753
1 vote
0 answers
310 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?
Felix Goldberg's user avatar
1 vote
0 answers
78 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 $A_{+}$ ...
Felix Goldberg's user avatar
0 votes
0 answers
86 views

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 ...
user76284's user avatar
  • 1,793
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65 views

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)=\...
Nicolas Medina Sanchez's user avatar
0 votes
0 answers
160 views

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 ...
Dong Ganzhe's user avatar
0 votes
0 answers
95 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 ...
Wieland's user avatar
  • 123