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
916
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A matrix that commutes with all symmetries of a vertex-transitive polytope
Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope.
Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\...
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How does one translate from convex hull to a set of facets (inequalities)? [duplicate]
Suppose I have defined a convex set as the convex hull of a set of points. (I know that all these points are "extremal points" of the convex set.) I know want to translate this description of the ...
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Polynomial size embeddings of toric varieties from polytopes?
Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan.
$X_P$ is always projective, because the collection of characters corresponding to the points $\...
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1
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Overlap count of convex combination of points [closed]
Let $x_1, ..., x_n\in\mathbb{R}^d$. We know that a point $x$ in the convex hull $\text{conv}(x_1, ..., x_n)$ may be expressed as convex combinations $x=\sum_{i=1}^n r_ix_i=\sum_{i=1}^n s_ix_i$ with ...
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An example of a "simple poset" which does not belong to a convex polytope
Are there examples of d-regular graphs (i.e. graphs where every node has exactly d adjacent nodes) which are not the 1-skeleton of a simple convex polytope?
UPDATE:
New version of the question: is ...
5
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Classification of vertex-transitive zonotopes
Zonotopes are convex polytopes that can be defined in several equivalent ways:
parallel projections of cubes,
Minkowsi sums of line segments,
only centrally symmetric faces,
...
I wonder whether ...
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167
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4-polytopes with only one kind of regular facet
Is there a neat way to show (or a reference that already proves) that
the 4-cube is the only convex 4-polytope in which all facets are regular 3-cubes?
the 24-cell is the only convex 4-polytope in ...
3
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1
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Are there any more polytopes whose 2-faces are identical 4-gons?
What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
$P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
all 2-faces of $P$ are ...
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Furthest distance half the diameter?
Let $S$ be the surface of a convex body, polyhedral or smooth,
embedded in $\mathbb{R}^3$.
For a point $x \in S$, let $F(x)$ be the set of furthest points
from $x$, measured by shortest paths on the ...
5
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1
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Iterated derivative and rectangular standard Young tableaux
We first make a few definitions, seemingly out of the blue (they are introduced/defined in this paper).
Let $F^0_{a}(z) = (1-z)^{-1}$ and define recursively
$$
F^{k+1}_{a}(z) = z^{a-1} \frac{d^a}{dz^...
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How wide is the Birkhoff Polytope?
This question is migrated from MSE where it turned out to be much harder than I thought. I still cannot figure this out. Does anyone have any ideas?
Define the width of a polytope $P \subset \mathbb ...
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Guises of the refined Eulerian numbers, generated by tangent vectors (OEIS A145271)
The Eulerian numbers (OEIS A008292, not to be confused with the Euler numbers) pop up in numerous scenarios in combinatorics and advanced analysis, one as the components of the h-vectors of the ...
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How can the same polytope have three different volumes? [closed]
I'm quite new to geometry and I came across the idea that the same convex polytope can have at least three different volumes.
Consider the permutohedron, formed by the convex hull of the n! points ...
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Gaussian mean width of normal random cones
Suppose $1 \leq n < m < \infty$ are integers. For $g \sim \mathcal N(0, I_n)$ define the gaussian mean width of a non-empty set $T \subseteq \mathbb R^n$ by
$$
w(T) := \mathbb E \sup_{x \in T} \...
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Partition complexity measure of the boolean cube?
Given $n$ points $p_1,\dots,p_n$ in $\{0,1\}^d$ my goal is to find $m$ index sets $\mathcal I_1,\dots,\mathcal I_m$ on the condition that each index set is a subset of $\{1,\dots,n\}$ on the ...
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Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?
I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope.
I am looking ...
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0
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Realizing 0/1-polytopes with shortest possible edge lengths
Has there been something written about the following question?
Question: Given a 0/1-polytope, what is the shortest edge lengths with which this polytope can be realized as a 0/1-polytope.
The ...
1
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1
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A source for $01$-polytopes
Can you recommend any books or survey articles on $01$-polytopes, thats is, polytopes with vertices in $\{0,1\}^n$?
I am less interested in random $01$-polytopes, but more in the combinatorial ...
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1
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The number of Hamiltonian circuits on a convex polytope embedded in $\mathbb{R}^N$
Recently I wondered whether there might be a natural topological complexity measure for convex polytopes embedded in $\mathbb{R}^N$. After some reflection it occurred to me that the number of distinct ...
2
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1
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Polytopes and polyhedral cones in complex Euclidean space
Given $A \in \mathsf{M}_{m \times n}(\mathbb{R})$ and $b \in \mathbb{R}^m$, the polyhedron with respect to $A$ and $b$, denoted by $P(A,b)$, is defined by
$$ \{ x \in \mathbb{R}^n \mid Ax \le b \}.$$
...
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Iterated polyhedron face twisting
Let $Q$ be a polygon in the plane. Modify $Q$ by rotating each edge about its
midpoint by $180^\circ$. The result is $Q$ again: No change.
This suggests exploring a similar operation in $\mathbb{R}^3$...
3
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1
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Regular triangulations of star-convex polyhedra with given boundary
Given an $n$-dimensional star-convex polyhedron $P\subset \mathbb{R}^n$ with simplicial facets, is it always possible to construct a regular triangulation $K$ of $P$ which does not subdivide the ...
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0
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Is there a simple polyhedral characterization of these integral points?
Given $n\in\mathbb Z_{>0}$ consider the set of $n$-tuples $$(a_1,\dots,a_n)\in\mathbb Z_{\geq0}^n$$ on following simple conditions
$0\leq a_i\leq 2^{2^n}-1$
If $a_i=b_{i,2^{n}-1}2^{2^{n}-1}+\dots+...
1
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1
answer
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Estimating volume of a simple object
Volume computation is $\#P$ hard.
Take the $[0,1]^n$ polytope.
Slice it by an half space inequality with $poly(n)$ bit rational coefficients into two unequal halves.
Volume of bigger section is $\...
6
votes
1
answer
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Edges of the contact polytope of the Leech lattice
Let $P\subset\Bbb R^{24}$ be the contact polytope of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda_{24}$.
Question: What are the edges of $P$?
Let'...
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1
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Given $H_{N}=\{\vec{x} \in [-1,1]^N:\sum_{i=1}^N x_i = 0\}$, what is the smallest subset $S \subset H_{2N}$ such that $\mathrm{conv}(S)=H_{2N}$
Motivation:
This is related to a different question I asked in April. It occurred to me while thinking about the sums of uniform random variables and it stuck in my mind because it's the special case ...
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Complexity of scissors congruence?
Where is the complexity of the problem 'Given two bounded compact convex integral polyhedra in $\mathbb R^n$ presented by $O(poly(n))$ integer valued halfspaces given by linear inequalities with ...
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0
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Two Questions on Tetrahedra and Platonic Solids [closed]
As was known to the ancients, two congruent regular tetrahedra can be inscribed in a cube and likewise 5 congruent regular tetrahedra can be inscribed in a regular dodecahedron. Is the converse to ...
5
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Sufficient criterion for a simplicial sphere to be polytopal
Are there any purely combinatorial criteria which allow you to deduce that a spherical simplicial complex is polytopal (i.e., there exists a simplicial polytope whose boundary is isomorphic to it)?
...
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Formula for exponential integral over a cone
While reading 'Computing the Volume, Counting Integral points, and Exponential Sums' by A. Barvinok (1993), I came across the following:
"Moreover, let $K$ be the conic hull of linearly independent ...
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Geometric realization of combinatorial self-duality in polytopes
Let's say I have a combinatorially self-dual polytope $P\subseteq\Bbb R^d$, i.e., its face lattice is isomorphic to its dual (you reverse the direction of the lattice order).
Question: Is it always ...
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2
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A conjecture concerning symmetric convex sets [closed]
Question:
Let's suppose that $S \subset \mathbb{R}^n$ is convex and symmetric so:
\begin{equation}
x \in S \iff -x \in S \tag{1}
\end{equation}
Now, if we define the radius of $S$ as $R$ such that:
\...
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0
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Projection of a polytope along 4 orthogonal axes
Consider the following problem:
Given an $\mathcal{H}$-polytope $P$ in $\mathbb{R}^d$ and $4$ orthogonal vectors $v_1, ..., v_4 \in \mathbb{R}^d$, compute the projection of $P$ to the subspace ...
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When the image of a convex set in $\mathbb{R}^n$ is still a convex set?
Here is tricky problem I came across when writing a paper. But I can't figure it out, so I ask for help here.
Let $M$ be a $n$-dimensional smooth manifold which is also a complete geodesic space. We ...
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0
answers
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Family of polytopes whose measure respects multiplication?
Is there a family $\mathcal{P}$ of integral polytopes and a polytope product $\star$ such that for every $n\in\mathbb N_{>1}$ $\exists p\in\mathcal{P}:vol(p)=n$ and
$\forall q\in\mathcal{P}\...
2
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0
answers
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Integrality of integral polytope?
Given a convex polytope with integral vertices presented in half-space description is there a way to test in
polynomial time or
polynomial hierarchy
if polytope has integral volume?
Finding ...
2
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0
answers
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Describing hull of vertex intersections of two convex bounded polytopes?
We have two convex bounded polytopes $P_1$ and $P_2$ where
a. $P_2\subseteq P_1$
b. $\mathcal{V}(P_2)\cap\mathcal{V}(P_1)\neq\emptyset$.
Is there a name for the polytope $P=\mbox{Conv}(\mathcal{V}(...
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1
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What importance does the Hirsch conjecture have to Simplex Complexity?
The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$.
After being open for decades, Francisco Santos has ...
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Alexandrov's generalization of Cauchy's rigidity theorem
Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions.
The relevant statement in the article is not linked to any source. The sources at the ...
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John's ellipsoid of a polytope
Suppose that $X$ is $\mathbb R^n$ with some polyhedral norm, that is, the unit ball of $X$ is an $n$-dimensional polytope. Assume that the John ellipsoid of $X$ is an Euclidean ball that touches every ...
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Does Banach-Mazur distance between regular polygons admit any structure that lends to approximation or exact results in particular situations?
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides. Do closed form or approximate results exist (at least at special infinitely ...
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Complexity of finding one vertex of a nonempty polytope
Let $P$ be a polytope given by some half-space description: $P=\{x\in\mathbb{R}^n: Ax\leq b\}$ for some $A\in\mathbb{R}^{m\times n}, b\in\mathbb{R}^m$, $m\geq n$. Assume that $x_0\in P$ for some given ...
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Upper bound on number of vertices in intersection (and union) of simplices
Let $S_1, \dots, S_k \subset \mathbb{R}^n$ be a set of (non-regular) simplices. Let $m_i$ indicate the number of vertices of simplex $S_i$ (we do not assume it is equal to $n-1$).
Is there a simple ...
2
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1
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Does quantifier elimination help here?
Suppose we have a quantified linear program
$$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$
$$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$
$$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$
$$...
2
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0
answers
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When do projection maps of polyhedra factor?
Given three polyhedra $P$, $Q$, and $R$ in dimensions $a$, $b$, and $c$ respectively, with $a\leq b\leq c$, with the additional condition that: $P=\pi_1(Q)=\pi_2(R)$, where $\pi_1$ and $\pi_2$ are ...
3
votes
1
answer
149
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A question about polytopes related to linear programming
Given linear functions $f_1({\bf x}),\dots,f_n({\bf x})$ on ${\bf R}^m$, let $K = \{(a_1,\dots,a_n) \in {\bf R}^n:$ the $n$ halfspaces $\{{\bf x}: f_i({\bf x}) \leq a_i\} $ have nonempty intersection$\...
3
votes
1
answer
286
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How to find the vertices of the set $\{v_i\in \mathbb{R}:a_1\ge v_1\ge v_2\ge \cdots\ge v_n\ge 0,\ q_2\le \sum_{i=1}^n p_iv_i\le q_1\}$
I am given a set of inequalities $v_1\ge v_2\ge \cdots\ge v_n\ge 0$, $q_2\le \sum_{i=1}^n p_iv_i\le q_1$, with $\{p_i\}_{i=1}^n,\ q_1,q_2$ positive reals, and only one bound for the coordinates: $v_1\...
3
votes
1
answer
271
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On decomposition of polytopes
Given $m$ number of convex polytopes each with $v$ vertices and described by $h$ hyperplane inequalities in $\mathbb R^t$ are there operations on these polytopes that combine then to give an $v^{\...
2
votes
0
answers
251
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Understanding the geometry of $H_{n}=\{\vec{x} \in [-N,N]^n:\sum_{i=1}^n x_i = 0\}$
I am not an expert in convex geometry but if we define $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{R}$ and $S_n = \sum_{i=1}^n a_i$ I suspect that for arbitrary $N \in [1, \infty) $:
...
1
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0
answers
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Mixed integer formulation of union of polytopes?
Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...