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Suppose that I want to plot the convex hull of the following set of vectors: namely, \begin{equation} Co(X)=Co\{(1,0,1,1),(0,0,2,1),(0,0,1,2),(0,1,1,1),(1,1,1,0),(1,1,0,1),(1,2,0,0)\} \end{equation} Because the matrix given by the original set of vectors has rank $ 4$ , its convex hull does not lie in a three-dimensional space. However, the matrix given by my original vectors can be translatedRead more

I’ve just been told in a comment of this question I posted two days ago that there exists a Matlab function that computes the centroid of any convex polytope in $ \mathbb{R}^n$ . The user who pointed out that Matlab function knows of no equivalence in Mathematica. Therefore, my question is: do we have anyRead more

Take $ a,\,b,\,c,\,d \in \mathbb R_+$ such that $ a+b+c+d=1$ . Define: \begin{equation} x_1 = \min(a+b,\,c+d)\,,\qquad x_2 = \min(a+c,\,b+d)\,,\qquad x_1 = \min(a+d,\,b+c)\;. \end{equation} I would like to prove that the set of all possible triples $ (x_1,\,x_2,\,x_3)$ , satisfying the previous condition, forms a convex polytope in $ \mathbb R^3$ . I am not sureRead more

I have a trouble with understanding the connection between polynomials and Newton polytopes. I will try to make a short introduction to my problem. Hope you will catch. In the end I will ask questions. I have a game with n players: $ 1,2,…,n$ . i-th player has $ d_i$ strategies and he allocates probabilitiesRead more

Let’s have a 10×3 matrix $ Ax≤b$ $ $ \begin{bmatrix} 0 & 1 & 1 \ 1 & 0 &1 \ 1 & 0 & 0\ 0 & 1 & 0\ 0 & 0 & -1 \ 0 & 0& 1\ 0 & -1& 0\ 0 & -1& 1\ -1 & 0& 0\ -1 &Read more