# Questions tagged [convex-cone]

This tag should be used with convex cones defined as usual within the context of Euclidean spaces or topological vector spaces, and their applications within geometry and topology, optimization, combinatorics and any other related area of mathematics using the classic definition of convex cone.

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### Polyhedral cone face lattice computation using a software. [closed]

I am looking for a software to compute the face lattice of a polyhedral cone other than Macaulay 2 and NORMALIZ.
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### What is the theory of the cones of minorant functions?

As a non - mathematician, I face some difficulty in understanding concepts such as convex cone, minorant/majorant function. The question is What is the theory of the cones of minorant functions? And ...
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### What is the distance between the centre of the sphere and the vertex of the cone?

A hollow right circular cone rests on a sphere as shown in the figure. The height of the cone is $4$ metres and the radius of the base is $1$ metre. The volume of the sphere is same as that of the ...
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### Defining the Polar set

For a subset P of R^n (real numbers) the polar set is defined by: $$P^*:= \{ y\in \Bbb R^n\mid y\cdot x \leq 1 \text{ for all } x\in P \}.$$ Can someone break the definition into plain english as ...
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### Difference of closed convex cones

I have read the following claim The difference of two closed convex cones in $\mathbb R^n$ can be non closed but I am not convinced and cannot manage to find a counter-example, can you find any? By ...
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### linear independence, convex hull, and cone

I am trying to show that for any set, $A\neq \{ 0\}$ and any $x\in co(cone A) \subset \mathbb{R^n}$, there exists linearly independent set $A' \subset A$ such that $x\in co (cone A')$. Here, co(A) ...
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### Given a closed, convex, full-dimensional cone $K$, how do I show that $u\in int(K) \iff u^tx>0 \quad \forall x\in K^*-\{0\}$?

Given a closed, convex, full-dimensional cone $K$, how do I show that $x\in int(K) \iff y^Tx>0 \quad \forall y\in K^*- \{0\}$ ? I've thought about applying the Hahn-Banach separation theorem ...
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### Probability that random variable is inside cone

Suppose $x\in\mathbb{R}^n$ is a random variable with mean $\mu$ and covariance $\Sigma$. Consider a stochastic convex optimization problem, i.e. an optimization problem with chance constraints, ...
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### is a positive ray is a convex cone?

A definition of a convex cone is given here, could anyone tell me whether a ray $C=\{(x,y): y=mx, x\ge 0,y \ge 0, m=\tan \theta, 0< \theta< 90\}$ on the positive quadrant satisfies the condition ...
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### $x \ge 0, z \ge 0, xz \ge y^2$ for symmetric $2 \times 2$ positive semidefinite matrix

I am told that the positive semidefinite cone in $\mathbf{S}^2$ is $$X = \begin{bmatrix} x & y \\ y & z \end{bmatrix} \in \mathbf{S}^2_+ \iff x \ge 0, \ \ \ z \ge 0, \ \ \ xz \ge y^2,$$ ...
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### Is the nonnegative orthant a “convex polyhedral cone”?

I am currently studying Convex Optimization by Boyd and Vandenberghe. Chapter 2.2.4 Polyhedra gives the following description of a nonnegative orthant: Example 2.4 The nonnegative orthant is the set ...
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### Mori cone: extremal ray intersections

On an algebraic surface, much can be said about the Mori cone, or cone of curves. In this question, I will be particularly interested in intersection properties. Several sweeping statements can be ...