# 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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### How can the same polytope have three different volumes?

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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### Relatively prime polytope extension complexity

What is the extension complexity of the $0/1$ vertexed polytope in $2d$ dimensions with property that integer represented by first $d$ coordinates is coprime to integer represented by second $d$ ...

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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 ...

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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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### 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 ...

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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 $\mathbb{R}^3$...

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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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### 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+...

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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 $\...

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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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### 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 ...