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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.

In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions $n$ as an $n$-dimensional polytope or $n$-polytope. Reference: Wikipedia.

For example a two-dimensional polygon is a $2$-polytope and a three-dimensional polyhedron is a $3$-polytope.

An important category of polytopes is the category of regular polytopes. These are the polytopes whose symmetry group acts transitively on its vertices, edges, faces, etc. In $2$ dimensions, these are the regular polygons, in $3$ dimensions, these are the Platonic solids and the Kepler-Poinsot polyhedra, in $4$ dimensions, these are one of six convex figures, or one of ten non-convex ones, and in higher dimensions, these include only analogs of tetrahedra, cubes, and octahedra.

826 questions
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Polyhedra vs Polytope

I am having a hard time understanding what is the main difference between a polyhedron and a polytope. Could anyone explain me what is the difference between these two structures?
user284639
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Fastest way to check existence of solution for a linear system of inequalities

What is the fastest way to check if there exists a solution to the inequality $A x \leq b$, with $A \in \mathbb R^{n \times m}$? I know this can be checked through the phase 1 of a linear programming algorithm, but I would like to know if there…
Tobia Marcucci
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In what kind of space does this object live?

Let me quickly build up some background. One way to build a hypercube is to take cubes, and start gluing them together, face to face, such that each edge is shared by $3$ cubes. You complete the hypercube with $8$ cubes. This involves rotating…
Josh B.
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Definition of simplex

From Wikipedia: an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. I was wondering if the definition is equivalent to say a simplex is synonym of a convex polytope? Is simplex defined only for…
Tim
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What's the algorithm of finding the convex combination of permutation matrices for a doubly stochastic matrix?

According to Birkhoff, $n$-by-$n$ stochastic matrices form a convex polytope whose extreme points are precisely the permutation matrices. It implies that any doubly stochastic matrix can be written as a convex combination of finitely many…
xzhu
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Meaning cone, ray, fan for polytopes

I'm trying to understand some mathematical operations and definitions for my project. Could you explain the meanings? $P_b =\{x∈\mathbb{R}^d : Ax≤b\}$ is a polytope. Let's have a 10x3 matrix $Ax≤b$ $$ \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 &1 \\ 1 & 0…
Soner from The Ottoman Empire
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Analytic center of convex polytope

I have a convex polytope defined by $Ax \leq b$. I want to know how to find the "analytic center" of my convex polytope, because my goal is to sample from the polytope using Markov chain Monte Carlo (MCMC) methods, and it mixes better if i start…
Andy Yao
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Standalone proof of a conditional part of Lagrange’s Four-Square Theorem?

Lagrange’s Four-Square Theorem — a special case of the Fermat-Cauchy Polygonal Number Theorem (FCPNT) — states that every natural number can be written as the sum of the squares of at most four integers. I’m trying to find a new proof of a…
Kieren MacMillan
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Where is the mass of a hypercube?

Question: Consider a hypercube $C := \{\vec{x}\in\mathbb{R}^n : \forall(i)\: |x_i|\leq 1\}$ and hypersphere $S_r := \{\vec{x}\in\mathbb{R}^n : |\vec{x}| = r\}$. Let $R(n)$ be the radius of the sphere for which the intersection $C\cap S_{R(n)}$…
Yly
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Do there exist uniform triangular prisms with all vertices in $\mathbb Z^3$?

It's quite easy to find a regular square prism (cube) or a regular triangular antiprism (octahedron) with vertices in $\mathbb Z^3$. Take for instance, take the convex hulls $$…
Peter Kagey
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Path-type description for (faces of) associahedra?

Recall that faces of associahedra are indexed by planar trees aka configurations of non-interesecting diagonals in polygons. And incidence corresponds to contracting edges / removing diagonals. Vertices of associahedra can also be indexed by Dyck…
Grigory M
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Can one embed $\frac{n^2(n^2-1)}{12}$ planes in $\Bbb R^n$ in a symmetric way to determine the Riemann curvature tensor?

To explain my question I need to discuss two notions from differential geometry (the Riemann curvature tensor and sectional curvature), but this is really a linear algebra / Euclidean geometry question, so I will provide background material. The…
Akiva Weinberger
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Projection of a hypercube along a long diagonal. What is this polytope called?

I was wondering what people call a certain type of shape. It is the shape formed by an orthogonal projection of a hypercube along one of its longest diagonals. In other words, fill in the missing entries: n-simplex : Triangle -> Tetrahedon …
Lucas
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Simplest Proof of the Six Regular 4-Polytopes

I'm not necessarily looking for a rigorous proof, more an outline of how an undergrad could count the regular 4D polytopes (and perhaps investigate what they look like) as explicably as possible.
Sup
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If $P$ is a polytope that is both simplicial and simple, then $P$ is a simplex or an $n$-gon

Question: Show that polytopes which are both simple and simplicial are either simplicies or two-dimensional $n$-gons. A polytope $P$ is simple if every vertex is contained in exactly $d$ facets. A polytope $P$ is simplicial if every face is a…
SparkyBoomMan36
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