All Questions
Tagged with co.combinatorics convex-polytopes
244 questions
1
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29
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Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
1
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0
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123
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Growth polynomial of the Associahedron graph ? (Is it approximately Gaussian ?)
Consider Associahedron, consider graph build from its vertices and edges. Choose some vertex. Let us count the number of vertices on distances $k$ from the selected vertex. Write a generating ...
- 24.9k
9
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144
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Which polytopes have compact realization spaces?
Let $P\subset\Bbb R^d$ be a convex polytope.
Its reduced realization space is the space of all combinatorially equivalent polytopes modulo projective transformations.
I am interested in polytopes for ...
- 13.6k
0
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0
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44
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Lattice points in the boundary of a Minkowski sum of two convex lattice polygons
Let $P$ and $Q$ be two convex lattice polygons in $\mathbb{R}_+^2$ and let $P+Q$ be their Minkowski sum. Given a set $S \subset \mathbb{R}^2$, we let $L(S) =\#( S \bigcap \mathbb{Z}^2)$.
The equality $...
- 183
0
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0
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81
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A generalized permutohedron as the sum of the dilatations of the faces of the standard simplex
I am trying to understand the proof of the statement, specifically it refers to a theorem stated by Postnikov in his text on permutohedra. So, this sentence claims the following:
If $\{Y_I \}$ is a ...
- 251
1
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1
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200
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Description of the generalized permutahedron
According to Postnikov, we know that the generalized permutahedron are describe as "polytopes obtained by moving vertices of the usual permutohedron so that directions of all edges are preserved&...
- 251
2
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0
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318
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What's the number of facets of a $d$-dimensional cyclic polytope?
A face of a convex polytope $P$ is defined as
$P$ itself, or
a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
- 43
4
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0
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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 ...
2
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0
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64
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Eulerian polynomial from Bruhat interval - h* of something?
Let $\sigma \in S_n$ be a fixed permutation.
Consider the polynomial
$$
P_{\sigma}(t) = \sum_{\substack{\pi \in S_n \\ \pi \leq \sigma}} t^{\textrm{des}(\pi)}
$$
where $\leq$ denotes Bruhat order, and ...
- 15.8k
4
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1
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181
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Denominators of rational polytopes in terms of hyperplane coefficients
Let $\mathcal{P}$ be a convex polytope in $\mathbb{R}^n$ given in the form $\mathcal{P} = \{ x \in \mathbb{R}^n\colon A x\leq b \}$. Suppose that the entries of $A$ and $b$ are integers. Then it is ...
- 24.2k
3
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124
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Does this construct Platonic solids?
Consider $\mathrm{O}(3)\curvearrowright\mathbb{R}^3$. Let $\Gamma\subseteq\mathrm{O}(3)$ be a finite group. Let $x\in \mathbb{R}^3\setminus\{0\}$ be a point such that $\mathrm{Stab}_\Gamma(x)$ has ...
- 930
4
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1
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298
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Does Kalai's $3^d$ conjecture hold for simplicial spheres?
Kalai's $3^d$ conjecture asserts that every centrally symmetric $d$-polytope has at least $3^d$ non-empty faces. This is open in general, but has been proven for simplicial polytopes.
Question: Does ...
- 13.6k
0
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90
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Which polytopes can be folded to an edge?
While playing with bar-and-joint linkages, I noticed that the skeleton of a regular 3-dimensional cube can be folded to a single edge (this can be achieved by first flexing the cube to bring it to a ...
- 1,305
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
6
votes
2
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291
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"Minimal" connected matroids
I'm interested in connected matroids $M$ on the ground set $[n]$ for which there is no connected matroid on $[n]$ of the same rank but with a strictly smaller set of bases (by inclusion). Equivalently,...
- 3,513
8
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2
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417
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Permutohedron and triangulation of cube via Eulerian numbers
The $h$-vector of the (simplicial complex given by the boundary of the polytope dual to the) permutohedron is the sequence of Eulerian numbers $A(n,k)=\#\{w\in S_n\colon \mathrm{des}(w)=k\}$.
Example: ...
- 24.2k
9
votes
1
answer
327
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The convex hull of Schur polynomial evaluations
Let $r\leq n$ and $d$ be positive integers. A probability vector is a vector of non-negative entries that sum to 1. For each probability vector $\lambda$ of length $n$, let
$$s(\lambda)=(\dim[\pi] \...
- 980
2
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0
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68
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Sampling theorems for partition polynomials (associahedra, noncrossing partitions / parking functions)
Define the associahedra partition polynomial
$$
\begin{split}
A(x) &= 1 + A_1(u_1) z + A_2(u_1,u_2) z^2 + A_3(u_1,u_2,u_3) z^3 + \cdots\\
& \qquad\qquad = 1 + \sum_{n \geq 1} A_n(u_1,...,u_n) ...
- 10.5k
4
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0
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287
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How can we prove this combinatorial identity?
This is a follow up on my earlier MO post. Let's recall the sets
$$\mathbf{K}_n=\{\mathbf{k}\in\mathbb{Z}^n: k_i\geq0, k_1+\cdots+ k_n=n, k_1+\cdots k_i\leq i, \text{for all $1\leq i\leq n$}\}$$
and $\...
- 43.2k
5
votes
1
answer
398
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Catalan sequences vs composition sequences
In the paper, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Pitman and Stanley studied the $n$-dimensional polytope
$$\Pi_n(\mathbf x)=\{y\in\...
- 43.2k
8
votes
1
answer
447
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Do this polyhedron and other set have names?
Updated: My first post had a mistake because I confused in my mind two different but related sets. Hopefully the description below is correct now.
Let $\Lambda$ be a finite set. Let $\Lambda^{(2)}$ be ...
2
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0
answers
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
3
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0
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187
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Approximating any $d$-dimensional convex shape that occupies a constant fraction of its bounding box with a polytope having $\mathrm{poly}(d)$ facets
Given any convex set $A\in\mathbb{R}^d$, we denote by $V(A)$ its $d$-volume. Furthermore, given any two convex sets $A_1,A_2\in\mathbb{R}^d$, we denote by $V_{A_1,A_2}$ the $d$-volume of the symmetric ...
- 1,677
7
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0
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162
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Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets
We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...
- 1,677
4
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2
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266
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A rational polytope that is not a 01-polytope?
A 01-polytope is the convex hull of some points $S\subseteq\{0,1\}^n$. I wonder, which polytopes can be represented (combinatorially) as 01-polytopes? There are polytopes that cannot have rational ...
- 13.6k
3
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1
answer
218
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Bounding the number of facets of a polytope to approximate a given convex shape in higher dimensions
We are given a convex shape $S$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume $V(S)$ of $S$ be $\tfrac12$ (I guess nothing changes for any other fixed ...
- 1,677
3
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0
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151
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Permutahedra Euler characteristic polynomials from cumulant-moment relation, a combinatorial proof?
Given the formal Taylor series, or e.g.f.,
$f(x) = 1 + \sum_{n \geq 1} m_n \; \frac{x^n}{n!}$,
the classical formal cumulants $c_n$ are generated from the formal moments $m_n$ via
$ \sum_{n \geq 1} ...
- 10.5k
1
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0
answers
154
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Volume of a polytope as its degenerates to be lower dimensional
Consider a polytope $P$ defined by the usual inequalities $A\mathbf{x}\leq \mathbf{b}$; let me assume that $P$ is not contained in a proper subspace. A result which I believe to true, but am not ...
- 44.7k
8
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1
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361
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Inscribed $n$-polytope with $2^n$ vertices of maximal volume
The question is in the title:
Question: Which inscribed $n$-dimensional polytope (inscribed in the unit sphere) with $2^n$ vertices has the largest possible volume?
Is it the $n$-dimensional cube? ...
- 283
2
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1
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137
views
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
9
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1
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889
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Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory
In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
- 10.5k
1
vote
1
answer
92
views
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
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0
answers
370
views
Combinatorial proof of a matrix equation
I'm looking for combinatorial proofs (using, e.g., trees) of the following particular matrix equation $(I)$ and also combinatoric operational analogs of its solution via matrix inversion and/or Cramer'...
- 10.5k
4
votes
1
answer
74
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Simplicial polytope with regular cones
Let $P$ be a convex simplicial polytope in $\mathbb{R}^n$. Can we find a convex simplicial polytope $P_0$ in $\mathbb{R}^n$ combinatorially equivalent to $P$, satisfying the following condition: The ...
- 405
3
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0
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179
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Polytope algebra and toric vareties
Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$.
For a simple polytope $P$ let us ...
- 21.8k
24
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1
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622
views
Polytope where each vertex belongs to all but two facets
Let $P$ be a (convex, bounded) polytope with the following property: for every vertex $v$, there are exactly two facets which do not contain $v$. Does it follow that $P$ is (combinatorially) a ...
- 4,653
1
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0
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83
views
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
1
vote
1
answer
101
views
Graph diameter of the omnitruncated $E_8$ polytope
What is the graph diameter of the 1-skeleton of the omnitruncate of the $E_8$ family of uniform 8-polytopes?
- 2,773
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
3
votes
1
answer
105
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What property of ranked poset ensures that it is determined by its vertex-facet incidences?
For a convex polytope, its face poset is combinatorially determined by vertex-facet incidences. Now suppose we have an arbitrary finite poset that is ranked, so I can still speak of vertices and ...
- 765
3
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0
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130
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Counting lattice polytopes by volume
For any $n \in \mathbb{N}$ and $B \in \mathbb{R}_{\geq 0}$, let $\mathcal{P}(n,B)$ be the set of $n$-dimensional convex polytopes $\Delta \subseteq \mathbb{R}^n$, taken up to integral, unimodular ...
- 1,116
3
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0
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111
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"Slim" directed polytopes: any established name for them?
This is a "looking for context" question.
Let's say that a polytope is directed if its 1-skeleton is an oriented graph with no cycles, one source, one sink. (Edit: let us additionally assume ...
- 765
2
votes
0
answers
54
views
Inverting "codimension matrix" for polytopes?
Let $P$ be an abstract polytope. Let's construct its square matrix $A$ as follows. Its lines and columns are labelled by all faces of $P$, of all dimensions. Put $A(F_1,F_2)=t^m$ if $F_1$ is a subface ...
- 765
3
votes
0
answers
154
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Ehrhart-Macdonald reciprocity with multiplicities
Let $P$ be a convex lattice polytope in $\mathbb{R}^n$. The function $L(t, P) = |\mathbb{Z}^n \cap t\cdot P|$ is a polynomial, and we have an equality
$$L(-t, P) = (-1)^nL(t, P^{int}),$$
where $P^{int}...
- 645
4
votes
1
answer
149
views
A combinatorial characterization of the central inversion of a polytope?
Given a convex full-dimensional polytope $P\subset\Bbb R^d$ (convex hull of finitely many points and not contained in any proper affine subspace) and a symmetry thereof (a linear map $\smash{T\in\...
- 13.6k
4
votes
0
answers
130
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"Baues poset" of shellings of simplicial polytope?
Let me start with some background I want to use as analogy.
Consider a (convex) polytope $P$ and its set of triangulations. Among all the triangulations, a well behaved subset are the regular ones: ...
- 24.2k
2
votes
1
answer
119
views
Anchor sets for lattice polygons: Part I
Suppose $V=\{(x_1,y_1), (x_2,y_2),\dots,(x_v,y_v)\}$ is a vertex set of lattice points satisfying
$$0=x_1<x_2<\dots<x_v \qquad \text{and} \qquad y_1>y_2>\cdots>y_{v-1}>y_v=0.$$
...
- 43.2k
3
votes
1
answer
336
views
Cohomology ring of a hypersurface in toric variety
Let $X^n$ be a smooth projective toric variety corresponding to a simple polytope $P$. It is well known that the cohomology ring $H^*(X)$ can be described in terms of combinatorics of faces of $P$.
...
- 21.8k
15
votes
2
answers
481
views
Ehrhart period collapse for $123\ldots k$-avoiding Birkhoff polytope?
For $1 \leq r \leq n$, let $\mathcal{B}^n_r$ denote the polytope of all real matrices
$$ \pi = \begin{pmatrix} \pi_{1,1} & \pi_{1,2} & \cdots & \pi_{1,n} \\
\pi_{2,1} & \ddots & \...
- 24.2k
4
votes
2
answers
190
views
Existence of a fundamental domain for the convex hull of group action on a rational polytope
Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}...
- 3,472