All Questions
Tagged with convex-polytopes convex-geometry
254 questions
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28
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How to calculate the vertices of a convex polytope (k-DOP)
I am currently reading Christer Ericson's Real-Time Collision Detection Book. The topic I'm particularly interested in, is the chapter about Discrete-orientation Polytopes (k-DOPs). In his words "...
- 1
0
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0
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37
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Constructing a minimum-volume outer approximation polytope with fewer facets
I am tackling the following problem:
Given a set of points $D \in \mathbb{R}^d$ and their convex hull, represented with $n$ facets, I want to construct a convex polytope $P$ with at most $m<n$ ...
- 101
6
votes
1
answer
347
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Is a ball the hardest body to approximate by polytopes (in the Banach–Mazur metric)?
$\DeclareMathOperator\conv{conv}\DeclareMathOperator\Vol{Vol}$In the paper "An extremal property of the hypersphere" by Macbeath, the following functionals were introduced (here $n$ is fixed,...
- 93
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72
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Existence of a sequence of $-1/1$-polytopes with certain geometric properties
Let $P_n \subset \mathbb{R}^n$ be a sequence of polytopes (A polytope is the convex hull of finitely many points). Let $B_n \subset \mathbb{R}^{n}$ denote the Euclidean unit ball. I am interested in ...
- 93
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0
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99
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Shortest loop through vertices of a convex polytope
Let $P$ be a convex polytope in Euclidean space $\mathbf{R}^3$ and $\Gamma$ be a closed curve which passes through all vertices of $P$. How small can the length $L$ of $\Gamma$ be? More specifically, ...
- 7,149
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69
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Degree of reflectional symmetry of (unbounded) convex polyhedra in Euclidean spaces
Let $U \subset \mathbb{R}^m$ be an open domain. I'm trying to come up with a measure of its degree of reflectional symmetry and I have a question. The post in two-part, where in PART I I introduce the ...
- 1,467
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0
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72
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Probability of being inside a convex n-dimensional polytop
I am currently conducting some post-grad research about wireless transmissions with uncertain transmission delays.
As part of the research, each individual transmission is modelled using a probability ...
0
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1
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133
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Centroid of Minkowski sum
Let $A$ and $B$ be two compact convex subsets of $\mathbb{R}^n, n\geq 2$. Assume $x_A$ and $x_B$ are their respective centroid. If we form the Minkowski sum $C=A+B = \{x+y\mid x\in A, y\in B\}$, what ...
- 161
4
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46
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Implementation of Friedman's algorithm of reconstructing simple polytopes
In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
0
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1
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103
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Convex sets via fixed point equations
I have an equation of the general form
$$ X = S \cup T X $$
where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear ...
- 877
1
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40
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Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces
Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
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7
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3
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703
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A continuous version of Carathéodory's convex hull theorem
A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...
- 7,149
0
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1
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172
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Density of the set of convex polygons in the Banach-Mazur distance
Is the set of convex polygons dense in the set of convex domains in $\mathbb{R}^2$, for the Banach-Mazur distance?
Any insight for a negative or positive answer is very much welcome!
- 191
2
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1
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159
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Conic hull of a rectangle
I have a simple question that appeared in research: For a rectangle $S :=[a_1,b_1] \times[a_2,b_2] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n$. Let $p_0 = (a_1,a_2,\dots,a_n)$, and define $p_i ...
- 275
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51
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Estimating the Hausdorff distance of parallel facets of convex polytopes
Background
Let $\mathcal{K}_P^n$ denote the class of open, convex, $n$-dimensional polytopes in $\mathbb{R}^n$ containing the origin. For each $K\in \mathcal{K}_P^n$, its gauge function $f_*:\mathbb{R}...
- 21
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1
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42
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A converse question about the polyhedrality under linear mapping
It is quite well known that for any polyhedral set $K$ and any linear mapping $A$, the set $AK$ is polyhedral and hence closed. I am more curious about the converse problem, namely:
Suppose $K$ is a ...
- 300
3
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1
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111
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Does a matroid base polytope contain its circumcenter?
Let $(X,\mathcal B)$ be a matroid on the ground set $X=(x_1,...,x_n)$ and with set of bases $\mathcal B$, and let $P\subset\Bbb R^n$ be its matroid base polytope (i.e. the convex hull of the ...
- 13.6k
1
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0
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163
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Membership test of convex set
Let $K$ be a compact convex subset of $R^n$ which has some positive gaussian measure, say at least 1/2. For each nonzero vector $u \in R^n$, we
define another compact convex set $K * u$ in the ...
- 11
4
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2
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217
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On faces of polytopes
$\newcommand\ext{\operatorname{ext}}\newcommand\R{\mathbb R}$Let $A$ be a convex polytope in $\R^n$ with nonempty interior.
Consider the closed convex cone
$$K_A:=\{(l,t)\in(\R^n)'\times\R\colon\, l(x)...
- 128k
1
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0
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61
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$\psi_2$ marginals of the permutahedron?
Let $K$ be a convex body.
I in particular care about the permutahedron.
I will view this as being the convex hull of all coordinate-wise permutations of the vector
$$v = \frac{1}{2n+2}(-n, -n+2,\dots, ...
- 2,183
4
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0
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52
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Quantifying error in the reconstruction of convex polytopes from moments
The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex ...
- 181
4
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1
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159
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Approximation of convex bodies by polytopes corresponding to smooth toric varieties
Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In ...
- 21.8k
1
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61
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Fitting a convex polytope with 𝑛 facets between two nested spheres
This is related to a research problem that is interested in approximation of spheres by convex polytopes.
Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...
1
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0
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36
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Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?
Suppose that $P$ is a polyhedron represented by
$$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$
and $P$ contains interior points. Moreover, the ...
- 33
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1
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101
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Maximum number of vectors with bounds on inner products (follow up question)
This is a follow-up question from my previous question.
Suppose there are (2n+1) vectors $\{m_1,m_2,...,m_n\}$, $\{p_1,p_2,...,p_n\}$ and $p^*$ in $R^{k+1}$. $m_i$ are weakly positive vectors. $p_i$ ...
- 23
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200
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Toric decomposition of multipartitions
Fix $k \in \mathbb Z_{>0}$. By a $k$-multipartition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $N$, I mean that each $\lambda_i$ is a partition of some $N_i$ and $\sum N_i = N$.
Let's call $\lambda$ ...
- 111
20
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433
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Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?
Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation
it stays a convex polytope,
it stays a combinatorial dodecahedron (i.e. its ...
- 13.6k
7
votes
1
answer
184
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In a set of n points on $R^d$, each point can be "well separated" from the rest by a linear functional. Is the dimension necessarily $\Omega(n)$?
For $x\in\mathbb{R}^d$ and $A\subset\mathbb{R}^d$, we say that $x$ is well separated from $A$ if there is a linear functional $f:\mathbb{R}^d\rightarrow\mathbb{R}$ such that $f(A)\subseteq [0,1]$ and $...
- 71
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224
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Characterization of curves contained in the boundary of convex bodies
Given a continuous closed curve $\gamma$ in $\mathbb R^n$ does there exist a convex body $K$ (convex set with non-empty interior) such that $\gamma\subset \partial K$?
I am looking for a reference to ...
2
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1
answer
308
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Intersection of the simplex with a linear subspace of codimension $2$
The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ is denoted $\mathring C$.
Let $S$ be the $n$-simplex:
$$S=\left\{x\in\mathbb{R}_+^n,\,\sum_{i=1}^n x_i=1\...
- 449
23
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1
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714
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Covering the unit sphere in $\mathbf{R}^n$ with $2n$ congruent disks
Let $v_i$ be $2n$ points in $\mathbf{R}^n$, with equal distance $|v_i|$ from the origin. Suppose that the convex hull of these points contains the unit ball. Is it known that $|v_i|\geq\sqrt{n}$? ...
- 7,149
1
vote
1
answer
119
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Volume ratio of polytopes with few vertices
The volume ratio of a convex body $K\subset \mathbb{R}^{n}$ is $v_r(K) = \inf_{\mathcal{E}\subset K} \left(\frac{Vol(K)}{Vol(\mathcal{E})}\right)^{1/n}$ where the infimum run over ellipsoids included ...
- 245
1
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1
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87
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Can upper bounds on totally monotone functions be taken (WLOG) to be themselves totally monotone?
Consider the following: fix a function $\bar{b} : \mathbf{R}_+ \to [0, \infty]$, and define
\begin{align}
\mathcal{S} \left( \bar{b} \right) := \left\{ b : \mathbf{R}_+ \to [0, \infty] \, \text{s.t.} \...
- 801
1
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0
answers
32
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Adjacent vertices of a permutohedron which is the orbit of a point $x$ with repeated coordinates
Question: Let $x=(x_1, ..., x_n) \in \mathbb{R}^n$, and let $P$ be the convex hull of the points formed by permuting the coordinates of $x$. Given a vertex $y$ of $P$, what is a general rule for ...
- 637
5
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2
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672
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Secondary polytope
Given a polytope $P$, what do the points of the secondary polytope correspond to?
I know that the vertices of the secondary polytope correspond to regular triangulations of $P$.
But what do the ...
- 43.2k
0
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0
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74
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Diameter of inner parallel body
Let's say I have a convex polytope $\mathcal{P} \subset \mathbb{R}^n$ with non-empty interior.
Let $\mathcal{P}=\bigcap_{i=1}^mH_i$ for some halfspaces $H_i$ and let $d=diam(\mathcal{P})=\sup\{\|x-y\|:...
- 93
0
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1
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101
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Estimation via projecting onto a convex body
Assume that $\theta$ is in a convex body $K \in \mathbb{R}^n$ and we observe $y = \theta + z$, where $z$ is a noise term (following, say, the normal distribution). Consider an estimator of $\theta$ by ...
- 773
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0
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118
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Weak derivative of projection onto probabilist's simplex
Let $\Delta_n:=\{x\in [0,1]^n:\boldsymbol{1}^{\top}x=1\}$ denote the probabilist's $n$-simplex and let $P:\mathbb{R}^n\rightarrow\Delta_n$ denote the (Euclidean) metric projection onto this simplex ...
- 5,405
1
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1
answer
139
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On the Lipschitz continuity of the unit-normal vector field of a polytope
Let $\mu$ be a probability measure on $\mathbb R^n$ and let $P$ be a compact polytope in $\mathbb R^n$. For any $x \in \mathbb R^n \setminus P$, let $p(x) \in P$ be (unique!) point in $P$ which is ...
- 6,853
2
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1
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113
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Existence of fine approximate of a convex body in $\mathbb R^d$ with convex hull of $\mathcal O(d)$ points
Let $K$ be a convex body in $\mathbb R^d$ which contains the origin and let $\theta \in (0,1)$.
Question. Is it always possible to find $n$ points $x_1,\dotsc,x_n \in \mathbb R^d$ such that
$$
\theta ...
- 6,853
2
votes
1
answer
137
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Optimal number of half-spaces in the $H$-representation of the convex hull of $n$ points in $\mathbb R^d$
Let $P$ be the polytope obtained as the convex hull of $n$ points in $\mathbb R^d$. This is the $V$-representation of $P$. Note that $P$ can also be represented as an intersection of closed half-...
- 6,853
1
vote
1
answer
92
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How to compute external angles of a hypersimplex?
Recently, I concern with the volume of the outer parallel body of a hypersimplex that is defined as follows
$$
\mathcal{H}_s(n,k)=\left\{ (x_1,\cdots,x_n):\sum_{i=1}^n x_i=k,x_i\in[0,1] \right\},
$$
...
- 550
1
vote
0
answers
65
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Minimum bounding rectangle of symmetric convex bodies in the plane : is the ball the worst case
The minimal rectangle containing the euclidean ball in the plane is the standard cube $B_\infty = [-1;1]^2$. I would like to know if the euclidean ball is the worst symmetric convex body to be ...
- 245
4
votes
1
answer
194
views
How to solve this minimax matrix optimization problem?
Recently, I want to know how well can a $\ell_1$ ball be approximated by the image of a $\ell_2$ ball under a linear transform. I formulate this problem as the following optimization problem.
\begin{...
- 550
0
votes
1
answer
175
views
Maximum vertex amount of low-dimensional simplex projection
Consider an arbitrary simplex $\mathcal{S} \subseteq \mathbb{R}^n$ ($\mathcal{S}$ is a polytope in $\mathbb{R}^n$ with $n+1$ vertices and non-empty interior). Let ${\bf P} \in \mathbb{R}^{m \times n}, ...
- 235
0
votes
1
answer
428
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The dimension of the normal cone of a face in a polytope
Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\
This seems to be intuitively obvious but I can't ...
2
votes
1
answer
133
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Discrete random walk on polytope via involutions
Let $P$ be a convex polytope (or more generally convex body, I suppose) in $V=\mathbb{R}^n$. For each $v\in \mathbb{P}V$, we define an involution $\tau_v\colon P\to P$ by setting $\tau_v(p)$ to be the ...
- 24.2k
1
vote
0
answers
83
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Weird transportation polytope
I'm looking to compute extremal points of a weird polytope. This polytope contains all matrices with positive entries $A \in \mathcal M_{n,m}\left(\mathbb R_+\right)$ such that:
every row sum except ...
- 686
0
votes
0
answers
165
views
Minimum circumscribed ellipsoid of $\mathcal H$-polytope
Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows
$$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$
I ...
- 235
8
votes
1
answer
153
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The polytope algebras generated by polytopes with rational vs arbitrary vertices
The polytope algebra was defined by P. McMullen in "The polytope algebra" Adv. Math. 78 (1989) as follows.
Let us denote by $\Pi'_\mathbb{R}$ the quotient of the free abelian group generated ...
- 21.8k