# Tagged Questions

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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### Seeking criteria for “threadable” pairs of centrosymmetric polyhedra

Let $A$ and $B$ be origin-centered centrosymmetric polyhedra in $\mathbb{R}^3$: "for every point $(x, y, z)$ [...] there is an indistinguishable point $(-x, -y, -z)$." Say that $A$ and $B$ are ...
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### Convex hulls of compact sets

Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
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### Lattice points in dilated polytopes and sumsets

Let $P$ be an integral polytope, that is, the convex hull of some points in $\mathbb{N}^d$. Let $p_1,\dots,p_m$ be all lattice points in $P$. Question: What is the condition on $P$ that guarantees ...
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### Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes. For a given $\delta$, let $n_\delta$ be the number of faces ...
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### Toric ideal of slice of a polytope?

Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...
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### Convex polyhedra jammed in $k$ disjoint holes

For a given convex polyhedron $P \subset \mathbb{R}^3$, I was imagining finding the optimal "fixing" of $P$ in holes (or jamming them in "mud"), which led to the following question. First, scale $P$ ...
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### Has this generalization of a determinant (assigning multiplicities to the rows) been studied?

I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant: Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
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### An integrality question about expressing an integer as a product of numbers below $n$

Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number, composed only of primes below $n$, and that can be expressed as $$N= \prod_{j=1}^{n} j^{x_j}$$ where $x_1$, $\ldots$, $x_n$...
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### Definition of self-dual polytope

Given a d-polytope $P$ we define the c-dual polytope as $P^\ast = \{y\in R^d \mid x\cdot y\geq -c, \forall x\in P\}$. Then I say that a polytope is c-polar self-dual if $P=P^\ast$. I cannot find this ...
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The formula for the j-function which employs polynomial invariants of the icosahedron, $$j(\tau)=-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5}$$ where, $$r^{-1}-... 1answer 119 views ### 2-neighborhood of a simplex Let \Delta be an n-1-simplex in {\mathbb R}^{n-1}. For each vertex v of \Delta let H_v be the hyperplane through v and parallel to the opposite facet. By 2-neighborhood of a simplex I ... 2answers 638 views ### “Derived” polyhedra and polytopes The notion of derived polygon is natural and leads to remarkable convergence. Start with a polygon, and replace it by locating a point on every edge a fraction \alpha between the two endpoints. For ... 0answers 152 views ### Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of \pi? After reading these very interesting questions, I came up with another one: Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ... 1answer 224 views ### Why do convex polytope options constrict with dimension, rather than expand? There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in \mathbb{R}^4, and then three regular polytopes in every dimension d > 4. There ... 1answer 166 views ### Convex deltahedra in higher dimensions There are eight convex polyhedra whose faces are equilateral triangles, so-called deltahedra: (Image from here) Q. Have the equivalent higher-dimensional ... 0answers 166 views ### realization spaces of 3-dimensional polytopes It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible. A proof of this theorem can be found for instance in ... 0answers 237 views ### Wasserstein distance, convex polytopes and extreme points Let us consider convex polytopes with K extreme points in \mathbb{R}^d. Let \mathbf{P} be such polytope, and let \text{ext}(\mathbf{P}) denote its set of extreme points, so that \mathbf{P} = \... 0answers 131 views ### Distance function from the origin to the boundary of a convex polytope Let K\subseteq \mathbb R^n, n\ge 3 be a convex bounded polytope containing the origin in its interior. Consider the unit sphere with geodesic distance (\mathcal S^{n-1},d) and define the ... 1answer 211 views ### n-dimensional Delaunay Triangulation of Lattices I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an n-dimensional lattice L and its Delaunay triangulation (partition of R^n into simplices ... 3answers 385 views ### Repeating an operation infinitely makes any convex n-gon a regular n-gon? For any convex n-gon P_{0,1}P_{0,2}\cdots P_{0,n}, let us consider the following operation : Operation : Let k=0,1,\cdots. Take n points P_{k+1,i}\ (i=1,2,\cdots,n) outside of n-gon P_{... 1answer 229 views ### A construction related to scissors congruence I was thinking about the following some time ago. My question is whether such things have been studied before. Let E_n be the abelian group with a generator for each (bounded) euclidean polytope of ... 2answers 656 views ### “MultiCatalan numbers” Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient$$ \binom{2k_1+3k_2+4k_3+...}{k_1+2k_2+3k_3+...,k_1,k_2,k_3,...} $$is ... 2answers 419 views ### Vertices of a Polytope Given two convex polytopes P,Q such that P\subset Q. We are given that all the vertices of P are also vertices of Q and all the facets defining planes of Q are also facets defining planes of ... 2answers 6k views ### The amplituhedron minus the physics Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it? All the descriptions I've so far encountered assume ... 2answers 262 views ### point in polytope Suppose I have the convex hull P of a finite collection of points in \mathbb{R}^d, and I want to see whether a point p is contained in P. This is a standard (some would say the standard linear ... 2answers 860 views ### Efficiently determine if convex hull contains the unit ball Given a set of n points in \mathbb{R}^d, is there an algorithm to determine if the convex hull contains the unit ball centered at the origin in polynomial time? The convex hull itself might have ... 1answer 112 views ### Estimates on the number of vertices of reflexive polytopes Suppose M \cong \mathbb{Z}^n is a rank n lattice, with dual lattice N. Suppose \Delta is a full dimensional lattice polytope (i.e. convex hull of finite lattice points) in M. Then \Delta ... 0answers 228 views ### Right-angled polytopes %This question is motivated by the little discussion here at the bottom. The following thing are known about hyperbolic right-angled polytopes: Compact hyperbolic right-angled polytopes do not ... 1answer 107 views ### Submodular measures on the hypercube By the hypercube I mean the lattice formed by all n-bit strings ordered by pointwise inequality. For example, 000 \leq 110, 010 \leq 110, 110 and 001 are not comparable. Further we have the ... 2answers 170 views ### name for a polytope constructed from a system of linear equations? To a system of inhomogeneous linear equations, one can associate a polytope, as follows. Let A\in\mathbb{R}^{m\times n}, b\in\mathbb{R}^m and V=\{x\mid Ax=b, \text{support of x minimal}\}\... 1answer 238 views ### n-simplex in an intersection of n balls Consider any n-simplex, n \geq 2. For each edge (i,j), consider n-ball B_{ij} such that vertices x_i and x_j are antipodal on this ball. Fix a point x_0 in the simplex. The question: ... 0answers 236 views ### Bi-spherical polyhedra Bicentric polygons have been studied: a polygon all of whose vertices lie on its circumcirle, and whose incircle is tangent to every edge: I have not been able to find a comparable literature ... 1answer 312 views ### Smallest regular simplex containing the unit cube in R^n What is the length e_n of the edge of the smallest n-dimensional regular simplex S_n containing the n-dimensional unit cube Q_n? In particular, is there n such that e_n<\sqrt{2}(n+... 0answers 176 views ### Tetrahedra passing through a hole Assume a plane P\subset\mathbb R^3 has a hole H, and that the hole is topologically a compact disc. Being so, P\setminus H does not separate the space. A regular tetrahedron \sigma^3 (of edge-... 2answers 253 views ### Hales's fan associated with a polyhedron In Hales's book (cited below), he associates what he calls a fan with any convex polyhedron in \mathbb{R}^3. I will not define his notion of fan, but let his figure (p.137) serve as a definition: &... 3answers 843 views ### On maximal regular polyhedra inscribed in a regular polyhedron Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1. For example, it's easy to prove that ... 2answers 234 views ### Do maximal polyhedra have algebraic volume? Is it possible to prove that for every n > 3 the maximal possible volume of a convex polyhedron having n vertices inscribed in a sphere of unit radius is an algebraic number? Update: What ... 1answer 168 views ### Equiprojective polyhedra Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open, and which some might find intriguing. Define an ... 2answers 343 views ### Covering convex polygons with inscribed disks The following problem came up when discussing mapping software (e.g., Google maps) with computer scientists. By B(c,r) I mean the planar disk (open or closed, it doesn't matter) of radius r around ... 3answers 240 views ### Cellular decomposition of a sphere to polytope My primary question is: given a cellular decomposition of a sphere is there any way to check if it can be embedded as the boundary of a polytope? My question is motivated by the following problem. I ... 0answers 143 views ### existence of lattice point in polytope This question was probably asked before but here goes. I have a convex polytope given by Ax\leq b for a specific integer matrix A and integer vector b. I need a simple method/result on how to ... 0answers 74 views ### Optimization over a variable domain defined as a convex hull of given points [closed] I have an optimization problem: \max_{\bf{x}} Z(\bf{x}), s.t. \bf{x} \in conv(\bf{S}) where \bf{x} is an n-dimensional vector, Z(\bf{x}) is a non-linear function. The domain of \bf{x} ... 2answers 242 views ### Volumes of convex vs non-convex polyhedra with prescribed facets areas It is known that given a set of Areas A_f and normals \vec{n}_f if \sum_f A_f \vec{n}_f=0 exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov ... 3answers 239 views ### How to determine if two rational cones intersect? Let$$\displaystyle C_1=C(r_1,...,r_{n_1})=( x\in \mathbb{R^d}|x=\sum_{i=1}^{n_1}\lambda_i r_i, \lambda_i\in \mathbb{R_{>0}})\displaystyle C_2=C(t_1,...,t_{n_2})=(x\in \mathbb{R^d}|x=\sum_{i=...
Let $E\subset \mathbb{R}^n$ be a cross-polytope: $$E= \left\lbrace x : \frac{|x_1|}{q_1}+\cdots+\frac{|x_n|}{q_n}\leq 1 \right\rbrace,$$ where $q_1,\dots,q_n$ are positive integers. I am interested ...