Questions tagged [polytopes]
In elementary geometry, a polytope is a geometric object with flat sides, which may exist in any general number of dimensions $n$ as an $n$-dimensional polytope or $n$-polytope.
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Probability of rolling a specified face for Archimedean solids
Here it is shown that (for a "suitable" mathematical definition of fairness) there are no fair $n$-sides die with odd $n$. The question originated with fairness being defined as, among other,...
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How many regular rhombic polyhedra exist?
Here are my definitions for regular, semi-regular, and irregular polyhedra:
A regular polyhedron is a convex, non-intersecting 3-dimensional shape made with polygon faces connected at edges and ...
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Why aren't there infinite star polytopes?
Surely we can assemble, from some schlafli symbol $\{p/q,n/m\}$ some arbitrary regular polytope? I understand that some of these constructions are infeasible, but surely not all of them are? Could we ...
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Properties of the "triangle inequality" polytope.
Consider the set of of all $n$-tuples of non-negative numbers $x_i$ summing to $1$ and such that $x_i+x_j \geq x_k$ for any triple of distinct $i,j,k$. This is closed subset of the simplex, cut out by ...
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Understanding changes in a polyhedron structure when there are perturbations on the restrictions vector
I have the polyhedron
$$ P := \left\{ {\bf x} \in \mathbb{R}^{\binom{n}{k}} : {\bf A} {\bf x} = {\bf b} , \hspace{0.3em} {\bf 0} \leq {\bf x} \leq {\bf 1} \right\} $$
where the matrix ${\bf A} \in \...
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Upper bound on the number of facets of a polytope
Let $P$ be a (simple, convex) polytope in $\mathbb{R}^n$ with at least $n + 1$ vertices.
Let $f_i$ be the number of elements with $i$ dimensions in $P$.
What is the maximum value that
$$f(P) = \frac{...
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Minimal representation of element in convex polyhedron
This is a refinement of the question Minimal representation of element in convex hull of $v_1,...,v_n\in\mathbb{R}^n$
Let $C$ a convex polyhedron in $\mathbb{R}^n$ and $v_1,..,v_k$ its extreme points ...
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symmetry of a polytope after mapping one facet
When doing homework on algebra, on the symmetries of regular polygons and regular polyhedra, I observed, that mapping vertices of a single edge in regular polygon to another or mapping vertices of a ...
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Align two 600 cellls with the E8 (Gosset $4_{21}$) polytope
It is just a few steps of algebra to show that the projections of vertices of an $E8$ polytope (Gosset polytope $4_{21}$ to the Coxeter plane can be brought in congruence with the projections of the ...
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Spherical excess in four dimensions
I would like to understand the tesselation of the four dimensional space by 24 cells from the point of view of spherical excess. Therefore, I tried to lift the ideas from three dimensions to four ...
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Parametrization of Polytope to Make it Full Dimensional
I have a question regarding polytopes. I have a constraint matrix and a vector. This matrix has a set amout of equalities on top and then the rest is inequalities. I know the number of equalities.
As ...
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Linear programming, sensitivity analysis and changes in the basic solution
I'm trying to understand how a change in the right-side vector b alters the optimal base. In particular, if I call $x^*(b)$ the solution to the following linear problem, depending on $b$ (and being ...
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Extreme point solutions to a generalized stable set LP
Consider the linear constraints:
$$ x_{ab} + x_{a'b'} \leq 1 \quad\forall a\neq a'\in A, b,b'\in B\\ x_{ab}\geq 0 \quad\forall a\in A, b\in B. $$
It is known that any vertex of this LP is half-...
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Is the Coxeter group $W_{n-1}=S_{n-1}$ the quotient of $S_n$ by a simple reflection?
I believe the following three formulations are equivalent:
Is the associahedron $A_{n-1}$ the quotient of $A_n$ by one of its largest faces?
Are the triangulations of an $n-1$-gon the quotient of ...
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Every point on the boundary of a polytope is on a face
I am a beginner to all this math involving linear programming and convexity. As I was going through the proof of the Fundamental Theorem of Linear Programming. The proof on Wikipedia (the entire thing ...
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