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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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### 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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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 $... 0answers 78 views ### Are “vector spaces” over a smooth scheme with constant fiber dimension locally free? I've got the follow question which drives me almost crazy as the answer seems to be simple. Given a morphism$p:V\to S$of schemes of finite type over some base field. Assume that$p$has all the ... 0answers 37 views ### Under which conditions is the union of conic hulls of sets in a cartesian product equal to$\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 ... 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 ... 2answers 127 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 ... 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, ... 1answer 112 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$R^2$is convex iff its (signed) curvature has a constant sign. I wonder if one can characterize smooth convex cones in$R^3$in a ... 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 ... 2answers 188 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? ... 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 ... 2answers 299 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 ... 2answers 145 views ### sensitivity analysis in conic optimization I have a conic optimization of the form:$\min_x \langle c, x \rangle$, 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 self ... 1answer 119 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 ... 1answer 586 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 ... 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$A_{+}$... 2answers 151 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 ... 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 ... 1answer 480 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 ... 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? 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\}...
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 $\mathbf{R}^{n}-0$....