# Tag Info

## New answers tagged convex-geometry

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### Is the minimum angle between a cone and a vector achieved on an extreme ray?

No. Consider $v=(0,0,-1)$ (yellow) and the two-dimensional cone with extreme rays $(1,1,0)$ (red) and $(1,-1,0)$ (blue). The angle is minimised for the cone element $(0,1,0)$ (green).
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### Proving the "centroid" property and the existence of corresponding convex polyhedron in Minkowski Problem

The first identity is a consequence of the fact that volume is translation invariant. For each each face $F_i$, let $h_i$ be $n_i\cdot x$ for any $x \in F_i$. The volume of the polyhedron is  \frac{...
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### Proving the "centroid" property and the existence of corresponding convex polyhedron in Minkowski Problem

The identity $\displaystyle \sum_{i=1}^k A_i \mathbf{n}_i = 0$ can be proven using the divergence theorem which states that the surface integral of a vector field $\mathbf{F}(x,y,z)$ over a closed ...
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### If a polyhedron's faces and vertex figures are convex, is the polyhedron convex?

The OP doesn't specify that the abstract polyhedron must be a finite set. So here's an infinite counter-example: a pentagrammic cylinder, or a stack of pentagrammic prisms, in hyperbolic space. Using ...
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### Convex optimization of order 2 polynomial

I realise the question is NP Hard(But i am still not sure so you may as well answer according to how you view it). since no one could answer, I will answer it myself. Take \$a_i \in \{0,1\} \forall i \...

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