**33**

votes

**14**answers

7k views

### Open problems in Euclidean geometry?

Which are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a ...

**1**

vote

**0**answers

81 views

### First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...

**-3**

votes

**0**answers

50 views

### Does it make sense to compare sets (polytopes) with different dimensions? [on hold]

In the context of integer programming, I am considering 3 different linear models for a given problem. The goal is to determine which formulation is the tightest, that is, the one whose polytope is ...

**4**

votes

**0**answers

64 views

### Rational $d$-simplices

Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$
such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational.
So a rational triangle has rational edge lengths and ...

**25**

votes

**2**answers

480 views

### Complete the following sequence: point, triangle, octahedron, . . . in a dg-category

Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish).
An object $X$ in $\mathcal C$ gives a "point":
$$X$$
A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...

**3**

votes

**0**answers

113 views

### Average nastiness of a Newton polytope

Given a Newton polytope $P$ inside the $d$-scaled simplex ( i.e $\alpha \in P \implies | \alpha |=d $ ), then we define the following quantity:
$$ P(x)= \left( \sum_{\alpha \in P} ...

**3**

votes

**0**answers

104 views

### Zeros of Hilbert series of affine toric varieties

Consider a convex rational polyhedral cone $C\subset\mathbb R^m$ with vertex at the origin. Let $X$ be the corresponding affine toric variety, i.e. $\mathbb C[X]=\mathbb C[\mathbb Z^m\cap C^\circ]$. ...

**6**

votes

**1**answer

299 views

### Regular unimodular triangulation for a certain simplex

Consider an $n$-simplex with vertices given by
$(0,0,\dots,r_i,r_{i+1},\dots,r_n)$ where $r_1,\dots,r_n$ are given natural numbers,
and $i=0,1,\dots,n+1$.
Does this simplex admit a regular, ...

**4**

votes

**1**answer

330 views

### Simplex in convex polytope, pulling triangulation

Let $P$ be a convex $d$-dimensional polytope.
I have two questions, related to triangulations of $P$.
Question 1:
Let $p$ be in the interior of $P$. Can I always find a triangulation of $P$,
such ...

**1**

vote

**1**answer

109 views

### Counting faces on multipermutahedra/multipermutohedra

A multipermutahedron is the convex hull of all permutations of a list of numbers. For example, $\Pi(0,1,2)$ generates a regular hexagon, and $\Pi(0,1,1,2)$ generates a cuboctahedron.
In general, ...

**8**

votes

**0**answers

98 views

### Tensor Product of Convex Sets?

I was wondering if such a concept was used anywhere. What I was thinking of is this. Consider two vectors spaces $V,W$ and convex sets $C_1 \subseteq V$ and $C_2 \subseteq W$ if we define $C_1 \otimes ...

**3**

votes

**0**answers

164 views

### An isoperimetric inequality for “order” polytopes

I am looking for an isoperimetric inequality for order-like polytopes.
An order polytope $K\in \mathbb{R}^n$ is defined by a set of linear inequaities:
$$ \forall i \; 0\leq x_i \leq 1 $$
and
$ ...

**12**

votes

**0**answers

99 views

### Rational inscribed realization of the regular dodecahedron

While it is clear that the regular dodecahedron $D$ cannot be realized with all integer coordinates, it is easy to find a polytope, which is combinatorially equivalent (face lattice isomorphic) to $D$ ...

**12**

votes

**0**answers

218 views

### Does there always exist a self dual polytope that contains a given polytope contained in its dual?

Suppose a polytope $P$ is contained in its dual polytope $\tilde{P}$. Does there always exist a polytope $Q$ that contains $P$ and is self dual $Q=\tilde{Q}$? Is there any bound on the minimal number ...

**7**

votes

**1**answer

198 views

### Partitioning a convex object without harming existing convex subsets

$C$ is a convex planar figure and $C_1,\dots,C_n$ are pairwise-disjoint convex subsets of $C$, like this:
A convex-preserving partition of $C$ is a partition $C=E_1\cup\dots\cup E_N$, , such that ...

**3**

votes

**0**answers

82 views

### regular triangulations of the product of two simplices

Is description of all regular triangluations of $\Delta^n\times \Delta^k$ known? (Regular triangulations are those which correspond to vertices of Gelfand--Kapranov--Zelevinsky secondary polytope, or, ...

**0**

votes

**0**answers

45 views

### Distinguishing (possibly lower dimensional) $1$-skeleton of a regular graph inscribed in a sphere

Consider you have two (possibly same) convex $1$-skeleton of a regular graph $A$ and $B$ in $m$-dimensions inscribed in a sphere with possibly exponential number of vertices in $n$-dimension with ...

**2**

votes

**2**answers

50 views

### Can we prove that a normal surface of an extreme point of a convex subset of a simplex is a separating hyperplane?

Let's assume that we have a simplex $G = \{x\in R^d|\sum_{i=1}^d x_i=1, x_i\ge 0 , i = 1, 2, .., d\}$ and a polyhedral convex subset $H \subseteq G$.
Is it possible to prove that for any extremal ...

**10**

votes

**2**answers

371 views

### “Database” of simplicial polytopes/spheres

Reading through various papers on polytopes I have come across really interesting examples of simplical polytopes and non-shellable (or non-PL) simplicial spheres but sometimes it is hard to keep ...

**12**

votes

**1**answer

237 views

### Covering the unit sphere by open hemispheres

Suppose $H_1,\ldots,H_{2n}$ are open hemispheres which cover $S^{n-1}$ with the property that removing any one of them leaves $S^{n-1}$ uncovered. Is it necessarily the case that the hemispheres can ...

**5**

votes

**1**answer

158 views

### Is every triangulation of a Euclidean ball by convex tetrahedra shellable?

Suppose you are given a 3-ball $B$ in $\mathbb{R}^3$ that is bounded by a PL sphere, a triangulation $T$ of $B$ by Euclidean tetrahedra. Is that triangulation necessarily shellable?
I know that if ...

**2**

votes

**1**answer

91 views

### Approximating Ehrhart Polynomial of Rational n-Tetrahedron

A set of positive integers $d_1, \dots, d_n$ describe a n-dimensional tetrahedron $T$ with the vertices
$$ (0,\dots,0), (1/d_1,0,\dots,0), (0, 1/d_2,\dots,0), \dots, (0,\dots,1/d_n).$$
Let $L_T(t)$ be ...

**0**

votes

**0**answers

55 views

### Quick way to compute Ehrhart polynomial of Young diagram posets?

Using the hook formula, it is easy to compute the volume of order polytopes obtained from posets with partition shape, since this is the same as the number of linear extensions.
To my knowledge, ...

**8**

votes

**0**answers

133 views

### Positivity of coefficients of Ehrhart polynomial of n-Tetrahedron

A set of positive integers $d_1, \dots, d_n$ describe two n-dimensional closed lattice tetrahedron:
$$ T = \left\{ (x_1, \dots, x_n) \in \mathbb{R}^n: \sum_{i=1}^n \frac{x_i}{d_i} \leq 1 \textrm{ and ...

**6**

votes

**1**answer

249 views

### What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem)

The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem.
Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether ...

**2**

votes

**1**answer

113 views

### Choosing the weights of a Voronoi diagram — is this function always the gradient of another function?

This question is related to the earlier question Weighted area of a Voronoi cell . As in that question, let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = ...

**12**

votes

**6**answers

2k views

### On the number of Archimedean solids

Does anyone know of any good resources for the proof of the number of Archimedean solids (also known as semiregular polyhedra)?
I have seen a couple of algebraic discussions but no true proof. Also, ...

**4**

votes

**1**answer

267 views

### What are the 4 convex simplicial 4-polytopes that have 6 vertices?

In Convex polytopes and related complexes by Klee and Kleinschmidt they state the number of $d$-polytopes with $d+2$ vertices is $\lfloor \frac{d^2}{4}\rfloor$.
I was wondering what the four ...

**5**

votes

**1**answer

133 views

### Maximal volume of a simplex inscribed in a spherical cap

Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\varepsilon)$ be the spherical cap with height $\varepsilon$ I am interested in the quantity
$$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in ...

**4**

votes

**1**answer

121 views

### what's the formula of the inradius of a general simplex? [closed]

As the title, I just want to know whether there is a general formula for calculating the inradius of a n-simplex. Thank you!

**1**

vote

**0**answers

37 views

### Is the complement of a vertex figure in an (abstract) polytope connected?

I consider an (abstract) regular polytope $P$, and $H$ a vertex figure of $P$. Is the complement $P \setminus H$ connected (as a poset, in the sense that its Hasse diagram, ignoring the improper ...

**10**

votes

**6**answers

1k views

### Database of polyhedra

As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, ...

**1**

vote

**1**answer

60 views

### Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces

Suppose you have two polytopes $P_1, P_2 \in \Bbb{R}^n$ given by
$$ P_1 = \lbrace x: A_1 x \le b_1\rbrace$$
$$ P_2 = \lbrace x: A_2 x \le b_2\rbrace $$
I wish to find their convex hull, that is a ...

**1**

vote

**0**answers

54 views

### Log convexity for the norm of a vector-valued function

Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory.
Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...

**6**

votes

**3**answers

214 views

### Basic question about polytope duals

The following must be well known. Is there a beginning or midlevel
text where the answer is discussed? Thanks.
Along with a polytope one has the notion of its dual which is officially
defined via ...

**2**

votes

**2**answers

123 views

### Number of facets of a polyhedron when a vertex is removed

My question, informally: I have a bounded polyhedron in R^n with k facets, and I want to remove a vertex of this problem. How many facets does the remaining polyhedron have at most?
More formally: ...

**7**

votes

**1**answer

128 views

### Classifying two-faces of four-polytopes

Motivation: This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to ...

**6**

votes

**2**answers

174 views

### Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...

**8**

votes

**1**answer

201 views

### Tighter Caratheodory on the moment curve?

The moment curve is the set of points of the form
$$(t,t^2,t^3,...,t^n) \in R^n$$
Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$.
...

**5**

votes

**1**answer

85 views

### Sample integer points of cross-polytope uniformly

For $r,d\in\mathbb{N}$, let
$$C_{r,d}=\{x\in\mathbb{Z}^d: \|x\|_1\le r\}\subset\mathbb{Z}^d$$
be the set of integer points of the $d$-dimensional cross-polytope with radius $r$.
What is ...

**4**

votes

**0**answers

117 views

### Reference for the notion of polyhedra “degenerations”

Let $P$ be a convex polyhedron and let $P(t)$ be a continuous deformation thereof, such that:
a) $P(0)=P$;
b) for all $t\in[0;1)$ the polyhedron $P(t)$ is strongly combinatorially equivalent to $P$ ...

**1**

vote

**2**answers

102 views

### An optimality condition for the corners of convex polytopes?

Let $H,H'\subset\mathbb{R}^m$ be two hyperplanes with unit normal-vector,
and let $P\subset\mathbb{R^m}$ be a convex polytope (defined via its corners $v_0, ... , v_n$, where $n\ge m$).
Let's ...

**28**

votes

**2**answers

1k views

### The logic of convex sets

Let me start with Helly's theorem: Let $A_1$, $A_2$, ..., $A_{n+2}$ be $n+2$ convex subsets of $\mathbb R^n$. If any $n+1$ of these subsets intersect (this means: have nonempty intersection), the so ...

**4**

votes

**2**answers

115 views

### Cluster Variables for non-convex n-gons

Most of the lectures and lecture notes on Cluster Algebras (at least from Combinatorial point of view) start with mutations of the diagonals of a convex n-gon (mostly the pentagon) as the illustration ...

**3**

votes

**1**answer

126 views

### The center of a minimal convex superbody

Is the following true?
CONJECTURE: $\,$ Let $\ B\ C\subseteq\mathbb R^n\ $ be convex bodies in $\mathbb R^n$ such that $\ C\ $ is centrally symmetric, $\ B\subseteq C,\ $ and $\ t\!\cdot\! B\ $ ...

**3**

votes

**2**answers

117 views

### How many different integer polytopes does square lattice have?

Let $E_n = \{ (i,j) : 0 \leq i,j \leq n-1 \}$. We say that a polytope $P$ is an integer polytope in $E_n$ if all vertices of $P$ belongs to $E_n$.
My question is how many different integer polytopes ...

**1**

vote

**0**answers

39 views

### Projecting on a convex compact polytope with special form

Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ...

**1**

vote

**0**answers

72 views

### Projecting on a a special polyhedron

Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron
$$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$
Notice that $\mathcal P_X$ is symmetric about the origin.
...

**1**

vote

**0**answers

37 views

### Checking integer points of given infinity norm on intersection

If we have convex region $\mathscr R$ given as intersection of a convex polytope $\mathscr P$ and an ellisoid $\mathscr E$ in $\ell^2$ norm is there an efficient way to test if there is an integer ...

**4**

votes

**0**answers

81 views

### Face structures of chain polytopes

For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain ...