# Questions tagged [dual-cone]

Use this tag for questions involving dual cones. In convex analysis, the "dual cone" to a set is the collection of all elements that form a "positive angle" with every element in the set. That is, given a set $S$ in a vector space $V$, we define the dual cone by $S^* = \{y:\langle x,y \rangle \geq 0 \text{ for all } x \in S\}$ (the precise meaning of $\langle \cdot,\cdot\rangle$ depends on the context).

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### Parameterizing the Dual Cone of a Cone defined by a Sublevel Set

While attempting to implement the results of a research paper, I've run across an interesting puzzle where I am seeking to identify an ellipsoidal set $\mathcal{C}$, but am only given the parameters ...
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### Epigraphical Cones, Fenchel Conjugates, and Duality

I'm trying to derive a result relating cones conceived as epigraphs of convex functions, duality, and Fenchel conjungates. Let me state exactly what I'm looking for: Let $\mathbb{E}$ be an Euclidean ...
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### Dual of conic program

Let $A$ be an $m \times n$ matrix (over $\mathbb{R}$), $b \in \mathbb{R}^m$, $c \in \mathbb{R}^n$ and $K \subseteq \mathbb{R}^n$ is a closed, convex, pointed cone with non-empty interior. We define a ...
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### Finding optimal savings with dual formulation (word problem)

We want to build a bridge over a river of width 2, with a pillar in the middle of the river. The bridge is symmetric and drops (linearly) to a minimum height of h meters below the initial level of ...
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### Why cones are represented by matrices

I see there are multiple definitions of cones: 1) Cone $K$ is defines as a set of vertices $[x_1, x_2, x_3, ...]$ with $$ as the base (starting point) 2) Cone $K$ is defined as intersection of ...
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### Minkowski sum of duals

I'm really struggling to prove the following statement: Let $\mathbb{E}$ be an Euclidean space, let $K,K_p,S\subseteq\mathbb{E}$ be a proper cone, a polyhedral cone and a subspace, respectively. If ...
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### Question about dual cone

Update: I have solved this problem, thanks to this inspiring post: Polar cone of the Polar cone of $K$ a closed convex cone is again $K$. I will add my solution later. ================================...
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### Is the nonnegative orthant isometric to itself under orthogonal mapping?

Problem description: (Informal). I simply want to know if there exists a necessary and (or at least) sufficient condition for an orthogonal matrix to map every point of the nonnegative orthant (say ...
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### subgroups of regular polytopes that preserve a given face

Say I have a regular polytope (e.g. it is vertex and face transitive). Given a face F, is it true that there are symmetry operations taking every vertex of F to every other that also send F to its ...
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### The extremal points of the dual polytopes of vertex transitive polytopes

Is the following true? I have a convex polytope $P$ that is vertex transitive - roughly speaking all extremal points of $P$ have the same face-sets (the polytopes are isogonal figures). It is known ...
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### Proof of closure, convex hull and minimal cone of dual set

In my studies of convexity I have recently come across the following: Let $V=\mathbb{R}^d ; d \geq 1$ be a Euclidean space and $S \neq \emptyset$ be a non empty set of the vector space $V$, ...
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### The dual of a circular cone

In my studies of cones and convexity I have recently come across the following unexplained piece of information presented: We consider a Euclidean space $R^d$ for $d \geq 1$ we look at the ...
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### Under what cases of cone $K$ is $x = \text{proj}_K(x) + \text{proj}_{-K^*}(x)$ possible for all $x$?

Here, $K^*$ is the dual cone of $K$: $K^* = \{x \mid x^Ty \geq 0 \forall y\in K\}.$ The property is true if $K$ is the nonnegative cone or the positive semidefinite cone. Does a more general ...
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### What mean T in formula?

I was reading a thesis (this one) when I came to some formulas with a T with came from nowhere. ,, . Could someone explain me what mean the T in these formulas ?
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