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Questions tagged [convex-polytopes]

Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming

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Minimum Euclidean squared norm in the convex hull of points with rational coordinates

This is probably known, but I have not located a reference. Let $P$ be the convex hull of $k$ points in $\mathbb R^n$ with rational coordinates. Consider the Euclidean square norm function $F:P\to\...
Claudio Gorodski's user avatar
5 votes
1 answer
264 views

Is there a polytope with an essentially unique shape?

More percisely: Question: Is there a (convex) polytope that has a unique realization up to, say, projective transformations? I suppose I have to assume that it has more than $d+2$ vertices/facets if ...
M. Winter's user avatar
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8 votes
1 answer
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Two questions on the permutohedron

The $n$-dimensional permutohedron $P_n$ is the polytope given by the convex hull of all the possible permutations of the vector $(1,2,\dots,n+1)\in\mathbb{R}^{n+1}$. So it has $(n+1)!$ vertexes. I ...
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3 votes
2 answers
438 views

If a polytope is centrally symmetric and combinatorially equivalent to a zonotope, is it a zonotope?

A zonotope is a polytope whose 2-faces are centrally symmetric. Question: If a polytope $P$ is centrally symmetric and combinatorially equivalent to a zonotope, is it itself a zonotope?
M. Winter's user avatar
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4 votes
2 answers
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Is there more than one pseudo-Catalan solid?

This question was asked on MSE a year ago. Motivation for this question can be found in other MSE questions here, here or here. Convex solids can have all sorts of symmetries: the platonic solids are ...
ARG's user avatar
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3 votes
1 answer
332 views

An example in symplectic geometry

$\DeclareMathOperator\SU{SU}$Let $M$ be a coadjoint orbit of dimension 6 of $\SU(3)$, and let $T$ be the maximal torus in $\SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map ...
Maria's user avatar
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3 votes
1 answer
123 views

Maximal number of visible vertices

Let $P$ be a three-dimensional convex polytope with $N$ faces; $O$ a point outside $P$. What is the maximal number $f(N)$ of vertices of $P$ which may be seen from $O$?
Dmitry Maximov's user avatar
3 votes
0 answers
135 views

Edges in the convex hull of the union of random polygons

Let $P$ and $Q$ be two convex polygons in $\mathbb{R}^2$. Given $a > 0$, denote by $aP$ its image under the dilation by $a$ centered around the origin (i.e. the polygon obtained by replacing each ...
Puraṭci Vinnani's user avatar
0 votes
1 answer
108 views

If $C_1\subseteq C_2$ are two closed convex cones that are pointed with $\partial C_1\subseteq \partial C_2$ then is $C_1=C_2$?

Let $C_1$ and $C_2$ be two proper full dimensional closed convex cones in $\mathbb{R}^n$ that are pointed. Suppose that $C_1\subseteq C_2$ and that the boundary of $C_1$ is contained in the boundary ...
slack tide's user avatar
4 votes
1 answer
218 views

Mass distributions for high dimensional simplex and cross polytope

In this question, it is shown that the radial mass distribution of an $n$-cube (i.e. the probability density for the distance from a point selected uniformly from within an $n$-cube to the cube's ...
Yly's user avatar
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1 vote
1 answer
409 views

Exact volume calculation of a polytope is NP hard under which restrictions?

Computing the exact volume of a polytope given in half space representation seems to be NP-hard. One paper I found proved it is hard for rational coefficients. (However, the paper itself was behind a ...
koedem's user avatar
  • 121
4 votes
0 answers
229 views

How to find the dimension of the polar cone of a convex cone generated by some given vectors

Suppose we have access to a generating set $\{v_1, ..., v_k\}\subseteq\mathbb{R}^n$ of the convex cone $C=cone(v_1, ..., v_k)$, where $cone(\cdot)$ is the conical hull (i.e. nonnegative span) of ...
Min Wu's user avatar
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11 votes
1 answer
652 views

How to correctly state Cauchy's rigidity theorem?

Cauchy's rigidity theorem is often stated briefly as Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent. As a more formal generalization to general ...
M. Winter's user avatar
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5 votes
0 answers
93 views

Which polytopes can be deformed while keeping their edge-lengths?

Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while keeping its combinatorial type, and keeping its ...
M. Winter's user avatar
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5 votes
2 answers
134 views

Is there a non-orthogonal linear deformation of a polytope that preserves edge-lengths and vertex-origin-distances?

Is there a polytope $P\subset\Bbb R^d$ (convex hull of finitely many points, not contained in a proper affine subspace), and a linear, but non-orthogonal transformation $T\in\mathrm{GL}(\Bbb R^d)\...
M. Winter's user avatar
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5 votes
0 answers
139 views

How does a map from permutahedra to associahedra factor through multiplihedra?

Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $...
Dasha Poliakova's user avatar
10 votes
3 answers
322 views

Integer decomposition property with a partial order

Let $\mathcal{P}$ be a convex lattice polytope in $\mathbb{R}^n$. We say that $\mathcal{P}$ has the integer decomposition property (or "is IDP") if for all $k\in \mathbb{N}$ and $\alpha \in ...
Sam Hopkins's user avatar
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3 votes
1 answer
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Solid angles at points in an orthosimplex

Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ ...
James Propp's user avatar
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19 votes
1 answer
928 views

Can every simple polytope be inscribed in a sphere?

It is known that not every convex polytope (even polyhedron, e.g. this one) can be made inscribed, that is, we cannot always move its vertices so that all vertices end up on a common sphere, and the ...
M. Winter's user avatar
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4 votes
3 answers
347 views

Minimal data required to determine a convex polytope

Let $P\subset \Bbb R^d$ be a convex polytope. Suppose that I know its combinatorial type (aka. the face-lattice), the length $\ell_i$ of each edge, and the distance $r_i$ of each vertex from the ...
M. Winter's user avatar
  • 13.6k
12 votes
1 answer
428 views

Curve with no embedding in a toric surface

I am looking for a smooth proper curve $C$ such that there does not exist any closed embedding $C \to S$ where $S$ is a (normal projective) toric surface. Since $C$ is smooth I believe it suffices to ...
Ben C's user avatar
  • 3,625
7 votes
1 answer
289 views

Separating a lattice simplex from a lattice polytope

Let $P\subset\mathbb{R}^n$ be a convex lattice polytope. Do there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ separating $\Delta$ from the convex ...
Puzzled's user avatar
  • 8,998
7 votes
0 answers
152 views

Probability of landing inside the convex hull of previously sampled points

Let $\{X_i\}_{0\leq i\leq\infty}$ be i.i.d. random vectors in $\mathbb{R^d}$. I would like to show that the probability of one point being in the convex hull of the others goes to one with the number ...
Maxim's user avatar
  • 233
2 votes
0 answers
47 views

Source request: Optimal bounds on signings of points from a convex body

I recently came across an old survey of problems in discrete geometry: https://pdfs.semanticscholar.org/c350/f4d4a9466fa6708d99ec1187c63d89bed20f.pdf Problem 2.1 from the list caught my eye. It states ...
Arun Jambulapati's user avatar
2 votes
1 answer
191 views

Polytope with most faces

Fix $m,n \in \mathbb{N}$ with $m \ge n+1$. Take $m$ points in general position in $\mathbb{R}^n$ and let $P$ be their convex hull. What is the maximal number of (external, codimension-one) faces that $...
zjs's user avatar
  • 465
1 vote
1 answer
184 views

Is the following generalization of the Caratheodory theorem true?

The colorful Carathéodory theorem (Bárány, 1982) considers $d+1$ "colors" $X_1,\ldots,X_{d+1}\subseteq \mathbb{R}^d$, and a point $x$ in the convex hull of each color ($x\in \text{conv}(X_i)$...
Erel Segal-Halevi's user avatar
2 votes
1 answer
295 views

Are orbit polytopes of rotation subgroup of Coxeter group combinatorially equivalent?

Suppose that $G\subset O(d)$ is a finite reflection (finite Coxeter) group. For any $v\in \mathbb{R}^d$ which is not fixed by any non-trivial $g\in G$, one can consider the orbit polytope (Coxeter) ...
Bob's user avatar
  • 439
3 votes
1 answer
159 views

Are cyclic orbitopes of permutahedra necessarily simplicies?

Suppose that $v=(v_1,\ldots, v_d)\in \mathbb{R}^d$ lies in the linear subspace $v_1+\cdots +v_d=0$, and moreover that the coordinates are pairwise distinct. The permutahedron \begin{equation} P(\...
Bob's user avatar
  • 439
9 votes
1 answer
262 views

Probability that random high dimensional vectors are all on the convex hull

Say I pick $n$ i.i.d. random standard normal points in $\mathbb{R}^d$. Roughly, as long as $n$ is much smaller than exponential in $d$, with high probability all points will be on the convex hull. ...
Geoffrey Irving's user avatar
1 vote
0 answers
38 views

Structural properties of polytopes for mainstream integer or linear programs

Are there any papers/textbooks/monographs that describe distinguishing properties of the polytopes that arise when solving the linear relaxation of well-known integer programs? For example, it is ...
Tom Solberg's user avatar
  • 4,049
4 votes
0 answers
82 views

Classification of space-filling (by identical copies) convex polyhedra in R^3

Is the classification of space-filling (by identical copies) convex polyhedra in R^3 is known ? There are only 5 "parallelohedra" - filling by translation. But if relax that property to ...
Alexander Chervov's user avatar
6 votes
0 answers
381 views

An inequality related to the numbers of faces of polytopes with d+2 facets

I would like to prove an inequality related to the number of $k$-faces of two $d$-polytopes with $d+2$ facets; see (1) below. Let $r>0$, $s>0$, $t\ge 0$, and $d\ge 2$ be such that $d=r+s+t$. We ...
Guillermo Pineda-Villavicencio's user avatar
0 votes
0 answers
109 views

The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool

In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...
user142929's user avatar
4 votes
0 answers
114 views

Can we combine the symmetries of two polytopes to create a more symmetric polytope?

Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$. The symmetry group $\mathrm{Aut}(P_i)\subset\...
M. Winter's user avatar
  • 13.6k
6 votes
1 answer
212 views

A polytope with congruent facets and an insphere that is not facet-transitive?

Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties? All facets are congruent, it has an ...
M. Winter's user avatar
  • 13.6k
9 votes
2 answers
321 views

Is a polytope that has in-spheres for faces of all dimensions already regular?

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points). A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent. So a 0-in-sphere ...
M. Winter's user avatar
  • 13.6k
3 votes
1 answer
84 views

Tilings of lattice polytopes by transformations of lattice polytopes

A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...
Display name's user avatar
5 votes
1 answer
361 views

What is known about the duals of cyclic polytopes?

What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)? In even dimensions, all facets of the dual are ...
M. Winter's user avatar
  • 13.6k
3 votes
1 answer
111 views

Reference for "every 5-dimensional polytope has a 3-gonal or 4-gonal face"

It seems to be folklore that every 5-dimensional convex polytope has a 3-gonal or 4-gonal face of dimension two. I was not able to track down a source for that claim. Alternatively, I would be ...
M. Winter's user avatar
  • 13.6k
5 votes
2 answers
304 views

Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?

The question is in the title: Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell? I consider only convex polytopes (convex ...
M. Winter's user avatar
  • 13.6k
1 vote
0 answers
116 views

Untruncate permutohedron of order 5

I would like to understand commutation classes of reduced expressions of the longest element in $S_5$ a little better. For this, it makes sense to look at the permutohedron of order 5. Since I am only ...
Bipolar Minds's user avatar
0 votes
0 answers
76 views

Visualization of higher Bruhat order B(5,2)

I made the following images of the higher Bruhat order B(5,2) (in the sense of Manin/Schechtman) with vZome: image 1 image 2 image 3 Unfortunately, in vZome its not possible do have regular octagons,...
Bipolar Minds's user avatar
0 votes
0 answers
53 views

Separability of Minkowski Sum of well-behaved sets

Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?
ABIM's user avatar
  • 5,405
4 votes
1 answer
200 views

Is there a spherical analogue of polar duality for spherical complexes?

Let $P$ be a spherical complex, which essentially means a tiling of a sphere, let us say the $(d-1)$-dimensional sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ to fix notation, where each cell is a ...
Malkoun's user avatar
  • 5,215
2 votes
0 answers
103 views

Polytopes with large dihedral angles

The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$. The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...
M. Winter's user avatar
  • 13.6k
2 votes
0 answers
53 views

Can a polytope with vertex-transitive edge graph or face lattice be made vertex-transitive?

Let $P\subset\Bbb R^d$ be a convex, full-dimensional polytope (convex hull of finitely many points, affine hull is the whole space), $G_P$ its edge graph and $\mathcal F_P$ its face lattice. Any of ...
M. Winter's user avatar
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3 votes
0 answers
115 views

Pyramids whose volume can be computed by simple cutting and glueing

Since this question remained without answers even after a bounty, I thought it might be time to ask it here. For which pyramid can you compute the volume from simple cut-and-glue processes? The Dehn ...
ARG's user avatar
  • 4,432
3 votes
0 answers
86 views

An exponential integral over a closed convex polytope

For any $T\geq 2$, let us define the polyhedron $S$ given by \begin{align*} S:=\{\underline{t}:=(t_0,t_1,t_{2},t_{3},t_{4},t_{5},t_{6},t_{7})\in [0,+\infty)^{8}:A\underline{t}\leq (\log T)\textbf{1}\} ...
The Number Theorist's user avatar
7 votes
1 answer
317 views

Minimum area of the convex hull of the union of a parallelogram and a triangle

This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of ...
Wlodek Kuperberg's user avatar
1 vote
0 answers
62 views

fast V representation update of polytope

Say that I have both the V and the H representation of a (possibly unbounded) polytope $P$. I want to append a some rows to the H representation, how can I quickly update the V representation to ...
user39430's user avatar
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