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4
votes
1answer
194 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 ...
3
votes
0answers
55 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 ...
3
votes
0answers
246 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
votes
0answers
79 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 ...
2
votes
0answers
84 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 ...
1
vote
0answers
57 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, ...
1
vote
0answers
74 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 ...
1
vote
0answers
84 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 ...
1
vote
0answers
113 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?
1
vote
0answers
63 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 $A_{+}$...
0
votes
0answers
37 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 ...