The polytopes tag has no wiki summary.
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Convex cone as sum of simplices?
In 3D a pyramid with a square base can be decomposed into the sum of two tetrahedra, i.e. two 3-simplices.
I am dealing with a homogeneous N-dimensional system of inequalities and my solution is a ...
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106 views
Groups acting on polytopes
I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs.
Their basic ...
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Sphere containment problem inside a rational convex polytope of general dimensions.
Given a positive number $r$ and a rational convex polytope (bounded polyhedra) described by its set of half-planes (system of linear inequalities: $A\cdot x \leq b$, where $A\in\mathbb{R}^{m\times ...
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Is the polar set of convex Polytope also Polytope
Let $P$ be a convex polytope.
How can I prove that the polar set of $P$ (lets call it $P^*$) is polytope?
where $P^*=\{x\in\mathcal R^n:\forall v\in P, |\langle x,v\rangle|\le1\}$ .
$Thanks$
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1answer
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How to prove a set of vectors $\{x\in \mathbb{R}^n: \| x\|<1\}$ is polytope?
Assuming I have a set of vectors $\{x\in \mathbb{R}^n: \| x\|<1\}$, how can I prove this is a polytope. What does it mean that the set is an intersection of a set of half-spaces? What properties ...
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Polytopes and matrices
I was on mirc and some user asked me this question, which I didn't know how to handle:
Let $P$ be the polytope of $m\times m$ real matrices, with elements in $[0,1]$ segment. and the sum of each row ...
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312 views
How to show the unit ball of the dual norm is also polytope?
Assuming the norm's unit ball is a convex polytope. How can one show that the unit ball of the dual's norm is convex polytope and/or polytope ?
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Find the vertices of the polytope
Let $x,n$ be 2 integers with $x<n$.
I need to find the vertices of the polytope $P$ of $2 \times n$ nonnegative matrices $A$ such that:
The first row in $A$ is summed to $x$.
$$\sum_{j=1}^n ...
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1answer
30 views
Part from “Regular polytopes” which I don't understand
This is a paragraph from "Regular polytopes" by Coxeter that I don't understand.
Although it is not always possible to include all the vertices of a polyhedron in a single chain of edges, it ...
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1answer
54 views
Removing redundant half-spaces that bound a convex polytope
I am computationally representing a convex polytope in $\mathbb{R}^n$ as a set $A$ of half-spaces that bound it; each such half-space is represented by a row vector $\mathbf{v} = \begin{bmatrix}v_1 ...
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Formula for finding integer points in fundamental parallelepipeds — reference for proof?
Does anyone know of a text where I can find a proof like the one found in Lemma 5.2 of here?
It's a formula for the integer points inside the fundamental parallelepiped of a simplicial integral cone. ...
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137 views
Explain `All polyhedrons are convex sets´
My teacher in course in Mat-2.3140 of Aalto University claims that 'All polyhedrons are convex sets' here. This premise was in a false-or-not-problem 'The feasible set of linear integer problem is ...
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Geometric Interpretation of $h_1(P)=f_{d-1}(P)-d$ for a polytope
In our lecture "Discrete Geometry 1", we are examining lineare realtions between the components of the f-vector and the h-vector of a polytope, in particular the Euler-Poincaré formula and the ...
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Are all polytopes also convex hulls?
It seems, at least in the 2-D case, that all polytopes are going to be convex. Does this hold if the dimensions are increased?
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Set of all affine maps between two polytopes
Let $V$ be a finite-dimensional real vector space, let $P, Q \subseteq V$ be polytopes with $P \subseteq Q$. (Let a polytope be defined as the convex hull of finitely many points.) I'm interested in ...
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55 views
Probability/Polytope concept
Let $X$ be a real random variable, with real values $X_i$ associated with probabilities $p_i \ge0, p_1 \le 1$, $i=1$ to $n$.
The variance $V_p(X)$, is, as usual:
$$V_p(X) = \sum^n_{i=1} p_i X_i^2 ...
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Existence of 5-d centrally symmetric self-dual polytope
Does there exist a 5-dimensional centrally symmetric self-dual polytope?
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relation between solution of a linear program and its perturbation
I have a linear program over a finite set of points $(x_1, x_2,\ldots, x_m)\in\mathbb{R}^n$:
$$
\max_j c' x_j
$$
Suppose the solution of this LP is obtained at a point $x_{j_1}$, which is a vertex ...
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244 views
Polytopes: proving completeness of set of facets
Let $P$ be a $d$-dimensional convex polytope.
$P$ is contained in $[0,1]^d$ and all vertices have only integral coefficients.
Given a set of facets of $P$, how to check that this set is maximal. i.e. ...
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Regular Polyhedrons
In $\mathbb{R}^3$, there are five regular polyhedrons (up to similarity), and can be parametrized by number of vertices, edges and faces. What is the number of regular polyhedrons in $\mathbb{R}^n$, ...
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Convex Polyhedron: How many corners maximum?
How many corners can a $n$-dimensinal convex polyhedron have at tops? Is it the same as the number of corners a $n$-dimensional simplex has?
EDIT:
By polyhedron $P$, I mean, that for some matrix $A ...
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The minimal face of a polytope containing a set
Encountered the following statement while reading a paper where it was stated without proof - am wondering why its true.
Suppose $P$ is a polytope, $M$ is a convex subset of $P$. Define $f(M)$ to be ...
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1answer
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Vertex set and combinatorial type of a constructed centrally symmetric polytope
Say P is an n-dimensional polytope in $\mathbb{R}^n$ with vertex set V, and c is a point in $\mathbb{R}^{n+1}$ which is not in $\mathbb{R}^n$. Let W = { c - v | v $\in$ V}, and P' = conv(V $\cup$ W).
...
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some of the properties of mixed volumes
Let $P,Q$ be polytopes in $\mathbb{R}^n$.
If $\lambda,\beta \geq 0$ are in $\mathbb{R}$, show that the $vol_n(\lambda P+ \beta Q)$ can be expressed in terms of mixed volumes as follows:
$\frac{1}{n!} ...
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mixed volume properties
Show that the mixed volume $MV_n(P_1,\dots,P_n)$ is invariant under all permutations of the $P_i.$
2.Show that the mixed volume is linear in each variable
$MV_n(P_1,\dots,\lambda P_i+\beta ...
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About the Platonic Solids in all dimensions
I am asking about the Platonic solids in all dimension, some reference about the proofs of many of the statement made in here.
I would like to here about how to think about higher dimensions mainly ...
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1answer
62 views
Convex Hull in Hierarchy Structure
As a beggining to convex hull algorithms lecturer introduced the structure which it's called "Hierarchy Structure".
Hierarchy Structure: in every given convex hull there is a maximum size convex hull ...
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1answer
91 views
Planar graph constructed from the edges of another planar graph
Let $G$ be a planar graph. We construct a graph $H$ from $G$ in the following manner :
The vertices of $H$ are interior points of the edges of $G$, one on each edge.
Two vertices of $H$ are joined ...
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1answer
192 views
$L_1$ projection of sum of convex functions onto polytopes
Suppose I have a function $f(x) : \mathbb R^n \to \mathbb R$ that is the sum of a given strictly convex function $g : \mathbb R \to \mathbb R$ in a single variable, i.e. $f(x) = g(x_1) + g(x_2) + ...
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Can i get every face of a polytope by taking a facet (of a facet (of a facet (…))) of the polytope?
Let $P$ be a polytope, i.e. a convex subset of a finite-dimensional real vector space with finitely many extreme points. Let $F$ be a proper face of the polytope, i.e. a subset $F \subset P$ such that ...
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107 views
a conjecture on norms and convex functions over polytopes
Suppose one has a convex, bounded polytope P $\subset R^n$ and a strictly convex function $f$ defined everywhere on $R^n$. $f$ has a unique minimum; and suppose this minimum occurs somewhere strictly ...
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A question about pyramids (polytopes)
Let's use the following definition of a face:
A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ ...
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Polytope as the intersection of closed half-spaces
I am stuck at a problem which looks very simple but which I still cannot prove.
Let $P$ be a $d$-polytope. Say that $P$ can be represented as the intersection of a given finite set of closed ...
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1answer
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Dual of a polytope — intersection of infinitely many halfspaces?
Let $S \subseteq \mathbb{R}^d$ be a $d$-dimensional convex set (i.e. $\exists d+1$ affinely independent points in $S$). Let the origin of the coordinate system lie in the interior of $S$ and let:
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Are there face lattice results for the following class of polytope constructions?
Let P be a d dimensional (convex) polytope, and Q a face of P. Let Q' be a translation of Q which is outside the affine hull of P (i.e., Q' contains a vertex not in the affine hull of P). Given the ...
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Geometric properties of a $d$-dimensional simplex in euclidean space
In school we have learned about objects in $2$-space and in $3$-space, with heavy emphasizes on the properties in $2$-space. My question can be formulated as follows:
What would we have learned in ...
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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.
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What is a cubical sphere?
Roughly, a cubical complex is like a simplicial complex except all the pieces glued together are combinatorial cubes of various dimensions.
A cubical sphere is a cubical complex that is homeomorphic ...
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How does one compute a bounding simplex for a set of points?
An algorithm I've implemented to tessellate an N-dimensional space requires starting with a bounding N-simplex.
Given a set of $k$ points $S_{0..k-1} \subset R^N$ is there a procedure to find a ...
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Alternative to Ziegler's “Lectures on Polytopes”
I am interested in alternatives to Ziegler's Lectures on Polytopes, which is the suggested textbook for a course I am attending. I find the conversational style of the book jarring.
