# 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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### The normal cone to a ball at a boundary point of the ball

I got stuck in a problem that in the end I needed to know what is the normal cone to a ball $B = \left\{ x \in \mathbb{R}^n : \lVert x \rVert \leq 1 \right\}$ (here $\lVert \cdot \rVert$ is an ...
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### Prove if the polyhedron $A x \leq b$ is bounded, then the cone hull of all rows of $A$ is $\mathbb{R}^d$

Let $P = \{x \in \mathbb{R}^d:Ax \leq b\}$ be a polyhedron, where $A = \begin{bmatrix} a_1^T\\a_2^T\\...\end{bmatrix}$, then the cone hull of $(a_1,a_2,\cdots)$ is $\mathbb{R}^d$. How to prove this ...
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### Basis of the intersection of a cone and its dual.

Let $a_1,\dots, a_m$ be vectors in $\mathbb R^n$, let $A\in\mathbb R^{n\times m}$ whose columns are $a_i$s. The cone generated by $A$ (denoted $\operatorname{cone}A$) is $C=\{ Aw: 0\leq w \}$, its ...
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### Does the closed convex hull of a compact set in the interior of a convex cone is still contained in the interior of the cone?

Let $C$ be a convex cone in a Banach space $X$ with nonempty interior. The set $A\subset {\rm Int}C$ is a compact subset, where ${\rm Int}C$ means the interior of $C$. Denote the closed convex hull of ...
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### Is conic programs with exponential cone solvable in polynomial time?

I'm trying to find out some theoretical guarantees of time complexity for my problems. My problem is to minimise a log-sum-exp function. I found that the minimisation of log-sum-exp function can be ...
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