Why is any subspace a convex cone? - Mathematics Stack Exchange
2015年2月28日 · Now why a subspace is a convex cone. Notice that, if we choose the coeficientes $\theta_1, \theta_2 \in \mathcal{R}_+$, we actually define a cone, and if the …
functional analysis - Convex cone generated by extreme rays ...
2017年7月26日 · To get sufficient criteria for the statement to be true, I guess one must resort to Krein–Milman type theorems (e.g. assume that the cone has a weakly compact base). For …
real analysis - When is a convex cone in $\mathbb{R}^n$ finitely ...
Counterexample in $\Bbb R^3$: the full cone with basis a disk (cone, for the layman). A finitely generated convex cone in $\Bbb R^3$ would just be some sort of full infinite pyramid with …
Why does a positive definite matrix defines a convex cone?
Now compute the quadratic form: $\vec{x}^\top A \vec{x} = x_1^2 + x_2^2$; which is definitely convex, since the square term is a convex function and the sum of two convex terms is …
real analysis - How to prove a cone is convex and closed?
2018年6月16日 · Proof of $(K_1+K_2)^* = K_1^*\cap K_2^*$: the dual of sum of convex cones is same to the intersection of duals of convex cones 1 A closed planar convex cone has at most …
Newest 'convex-cone' Questions - Mathematics Stack Exchange
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, …
Is a convex cone a convex polyhedron? - Mathematics Stack …
2016年4月13日 · EDIT: Just in order to avoid confusion, the definition of convex polyhedron I am thinking of is: the set of solutions to a finite number of linear equalities and inequalites. convex …
geometry - Faces of convex cones - Mathematics Stack Exchange
2021年6月16日 · Let $K\subset \mathbb{R}^n$ be a closed, convex, pointed cone and $\text{dim}K=n$. A convex cone $F\subset K$ is called a face if $F=K\cap H$, where $H$ is a ...
Dual of a dual cone - Mathematics Stack Exchange
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their …
Is the nonnegative orthant a "convex polyhedral cone"?
2020年1月31日 · 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 …