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$C\cap (-C)$, where $C\subseteq \mathbf{R}^n$ is a closed convex cone

That is wrong, take for instance $C=[0,\infty[\times\mathbb R$, then $C\cap -C = \{ 0\} \times \mathbb R$. In general $C\cap -C$ is the largest vector space contained in $C$ (and is empty if $0\notin ...
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Image of a rank-2 matrix by a nonlinearity

We can always construct an $M$ such that the rank of $\sigma(M)$ is at least as large as $\min\{n_1,n_2\}-1$. For instance, suppose $n_1\le n_2$. Let $v^T=(n_1,n_1-1,n_1-2,\ldots,1,0)$ and $w^T=\...
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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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How to show $\partial f(x) =\{\nabla f(x) \}$ for a convex function?

$\text{First part of the proof is simple. I prove the second part. Suppose d is a vector of norm 1 and u} \\ \text{is a subgradient of f at x which is not} \ c \nabla f(x) \ \text{for some constant c (...
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Interpretation of a formula for convex combinations

If we divide through by 2, we get $$\frac12 (\lambda^i a_i + \mu^j b_j) = \lambda^i\mu^j \frac{a_i + b_j}2.$$ In words, the midpoint of {a convex combination of the $a_i$} and {a convex combination of ...
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Upper-bound on volume of polytope inscribed in the sphere

The following is a direct consequence of Theorem 1 in Elekes. "A geometric inequality and the complexity of computing volume." Discrete & Computational Geometry (1986). If $k_n\in o(2^...
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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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