All Questions
4,825 questions
10
votes
2
answers
254
views
Is the face lattice of the cube a polytope graph?
The face lattice of a
convex polytope $P\subset\Bbb R^d$ is the partially ordered set whose elements are the faces of $P$ ordered by inclusion. We can turn it into a graph by considering its Hasse ...
- 13.6k
1
vote
1
answer
181
views
Relation between the root lattice of $\mathrm{SO}(7)$ and the root lattice of $G_2$
The root lattice of $\mathfrak{so}(7)$ is given by the following 18 roots:
$$
\left(\begin{array}{c}0\\0\\1\end{array}\right)
,
\left(\begin{array}{c}0\\0\\-1\end{array}\right)
,
\left(...
- 369
1
vote
0
answers
100
views
Perfect 'cuboiding' of cubes and cuboids
We try to add a bit to ref 2 listed below. In this post, by 'cuboid', we mean only rectangular cuboids - hexahedra with all faces rectangles and adjacent faces meeting only at right angles. A special ...
- 5,979
0
votes
0
answers
42
views
Regularization of the Laplacian on $\mathbb{R}^d$ and approximation schemes
this question is somewhat naive, but I am trying to understand the meaning of the regularized resolvant of the Laplacian on $\mathbb{R}^d$, and how it relates to a discrete approximation. Particularly,...
- 841
4
votes
1
answer
438
views
Perfect squaring of rectangles
A perfect squaring of a rectangle may be defined as a partition of the rectangle into finitely many squares all of which are mutually non-congruent. https://en.wikipedia.org/wiki/Squaring_the_square ...
- 5,979
3
votes
1
answer
239
views
The realization space of non-convex polyhedra - What is known?
The space $\mathfrak R_{\mathrm c}(P)$ of convex realizations of a (3-dimensional, spherical) polyhedron $P$ is known to be well-behaved: it is a contractible manifold of dimension $\#\text{edges}+6$ (...
- 13.6k
0
votes
0
answers
28
views
Example of a matrix -HDH that is not PSD (with non-euclidean distances D)
It's widely known that, given a matrix of squared Euclidean distances, $\mathbf{D}_{ij} = \| \mathbf{X}_i - \mathbf{X}_j \|^2$, and the centering matrix $\mathbf{H} = \mathbf{I} - \dfrac{1}{n}11^T$, ...
- 101
1
vote
0
answers
86
views
Busemann-Petty type problems on complex vector spaces [closed]
We recently published an article on Busemann-Petty type problems (see https://arxiv.org/abs/2404.05630). As we experienced several times that as soon as an article is published, no updates/corrections/...
0
votes
0
answers
41
views
Trying to extend a theorem on Tiling with mutually non-congruent triangles
In the light of Cubing the cube - as 'perfectly' as possible, We try to slightly 'relax' the main theorem proved by Kupaavski, Pach and Tardos here:
https://arxiv.org/pdf/1711.04504.pdf
...
- 5,979
9
votes
0
answers
186
views
Cubing the cube - as 'perfectly' as possible
Ref: https://en.wikipedia.org/wiki/Squaring_the_square
A perfect cubing of a cube is a partition of the cube into some finite number of smaller cubes that are pair-wise non-congruent. The above page ...
- 5,979
1
vote
0
answers
95
views
Distance between two convex sets
Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
- 125
0
votes
0
answers
76
views
Wasserstein space isomorphic to original space?
Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$?
Note that there is a canonical non-...
8
votes
4
answers
927
views
What are the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?
I asked this question on MSE here.
Given a quadrilateral with side lengths $a,b,c$ and $d$ (listed in order around the perimeter), t's known that the area, is always less than or equal to $\frac{(a+...
- 541
6
votes
1
answer
413
views
How many unit cubes are needed to 'hide' a unit cube fully in 3D?
Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...
- 5,979
3
votes
1
answer
199
views
Product of low dimensional Hausdorff measures
Let $\mathcal{H}^n$ and $\mathcal{H}^m$ be Hausdorff measures on $\mathbb{R}^n$ and $\mathbb{R}^m$. We know that the product measure $\mathcal{H}^n\otimes \mathcal{H}^m$ is the Hausdorff measure $\...
- 73
2
votes
1
answer
154
views
Continuity of the volume function
Consider a continuous map $F:(a,b)\times\mathbb{S}^n\to\mathbb{R}^{n+1}$ such that for any $t\in(a,b)$, the map $F(t,\cdot)=F_t:\mathbb{S}^n\to\mathbb{R}^{n+1}$ is Lipschitz continuous. The $n$-...
- 73
1
vote
0
answers
128
views
Sum of upper semi continuous and lower semi continuous functions
Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
- 1,043
0
votes
0
answers
197
views
Find which section of a convex polytope a point belongs to
Consider the convex polytope in dimension $n$ with vertex set $V$ given by the origin and the $n$ points
$$
e_i=\begin{bmatrix}0,\dots,0,\underset{i\text{-th coordinate}}{1},0,\dots,0\end{bmatrix}, i\...
30
votes
2
answers
2k
views
Packing an upwards equilateral triangle efficiently by downwards equilateral triangles
Consider the problem of packing an upwards-pointing unit equilateral triangle "efficiently" by downwards-pointing equilateral triangles, where "efficiently" means that there is ...
- 114k
6
votes
0
answers
197
views
What are compact manifolds such that GROWTH (of spheres volumes) is well approximated by the Gaussian normal distribution?
Consider some compact Riemannian manifold $M$. Fix some point $p$.
Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$.
Consider growth function $g(r)$ to be ...
- 24.9k
5
votes
0
answers
127
views
Does the permutohedron satisfy any minimal distortion property for graph metric vs Euclidean distance?
We can look on the permutohedron as a kind of "embedding" of the Cayley graph of $S_n$ to the Euclidean space. (That Cayley graph is constructed by the standard generators, i.e. ...
- 24.9k
6
votes
2
answers
189
views
Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon
For any point P in the interior of a convex polygon, the sum of the angles subtended by the edges of the polygon is obviously 2π.
Given a convex polygon, how does one algorithmically find the point (...
- 5,979
3
votes
0
answers
109
views
What Cayley graphs arise as nodes+edges from "nice" polytopes and when are these polytopes convex?
The Permutohedron is a remarkable convex polytope in $R^n$, such that its nodes are indexed by permutations and edges correspond to the Cayley graph of $S_n$ with respect to the standard generators, i....
- 24.9k
3
votes
1
answer
156
views
Concentration of measure on spheres with respect to a unitary of trace approximately zero
Cross-posted from MSE, where it hasn’t received any answer yet:
This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
- 2,830
6
votes
1
answer
207
views
Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
- 101
2
votes
0
answers
64
views
Limiting distribution of separated points in a unit square
Let $n$ and $r$ be fixed, and consider the following process, with $S=\emptyset$ to start:
For $i\in\{1,\dots,n\}$:
Sample a random point $X$ in the unit square.
If $X$ is a distance at least $r$ ...
- 4,049
1
vote
1
answer
84
views
Simple convergence of convex compact set implies Hausdorff convergence
I am wondering about the following :
In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
- 125
9
votes
2
answers
470
views
Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
Prove the inequality
$$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$
where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
- 99
4
votes
0
answers
46
views
Implementation of Friedman's algorithm of reconstructing simple polytopes
In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
1
vote
1
answer
172
views
Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?
I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them ...
- 1,465
5
votes
1
answer
311
views
Quotients in categories of metric spaces
There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
- 16.4k
22
votes
1
answer
1k
views
Does greedy circle packing exhaust the measure of every bounded open set in the plane?
The greedy circle packing of a bounded region in the plane is the result of placing at each stage the largest possible disk into the region that remains uncovered.
The greedy circle packing of a ...
- 236k
10
votes
1
answer
355
views
Is the group of translations of an affine plane always commutative?
$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, ...
- 41.8k
1
vote
0
answers
40
views
Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces
Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
- 5,979
2
votes
0
answers
244
views
Convergence of metric and eigenvalues on a tubular neighbourhood
Background:
Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
- 537
4
votes
0
answers
133
views
Curiosity about "conditional trig identities"
Perhaps this should be cross-posted on Math Stackexchange, but it came up in the context of some research mathematics (quaternion orders, etc.) In this context, I have three angles $\alpha, \beta, \...
- 13.3k
13
votes
0
answers
378
views
Is a convex polyhedron determined by its edge lengths and angular defects?
Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$.
The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$.
Question:
Is a ...
- 13.6k
-2
votes
1
answer
62
views
Inner Products of Elements in Spherical Cap [closed]
I am interested in understanding what is the lowerbound on the inner products of two elements of a sphere. Based on my intuition in dimension 2, I come up with the following conjecture. I appreciate ...
- 139
2
votes
0
answers
61
views
For riemannian manifolds, how close can a mapping from atlas be to an isometry?
Let $(M, g)$ be an $n$-dimensional $C^k$ (or $C^\infty$) Riemannian manifold. On $M$ we can define metric $d_g$ as the infimum of lengths of curves that connect given two points.
Fix $x \in M$ and $r&...
- 820
9
votes
0
answers
251
views
Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces
Is it know which metric spaces $M$ do have the following property: there is $\varepsilon>0$ and a maximal $1$-separated set which is $(1-\varepsilon)$-dense?
In other words, when does at set $S\...
- 799
3
votes
0
answers
226
views
Are 1-Wasserstein and 2-Wasserstein distances between multivariate normal distributions equivalent?
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$W^p_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
- 1,048
10
votes
3
answers
460
views
Do triple-linked graphs exist?
Lets say that a finite simple graph $G$ is (intrinsically) fully triple-linked if for each embedding of $G$ into $\Bbb R^3$ we can find three disjoint cycles $C_1,C_2,C_3\subset G$ whose embeddings ...
- 13.6k
0
votes
0
answers
39
views
Max-flow modeling with unified vehicle and commodity variables
I am working on a network flow problem that involves routing through a time-space network. The network consists of:
A single source node and a single demand node.
A fleet of vehicles with specified ...
- 101
4
votes
1
answer
244
views
Reference request: Fréchet embedding
Given a separable metric space $(X,d)$, we have an isometric embedding $\iota:X\to\ell^\infty$ given by taking $(x_n) _{n \ge 0}$ to be the countable dense subset and sending $\iota(x)_n=(d(x,x_n) - d(...
- 105
1
vote
1
answer
127
views
Connectedness of fibers of almost Riemannian submersions
EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary.
Given $f\colon M\to N$ be an $\varepsilon$-...
- 21.8k
1
vote
0
answers
65
views
Algorithm to generate configurations with kissing number 12
That the kissing number of a sphere in dimension 3 is 12 is well known. However, it is also known that there is a lot of empty space between the 12 spheres. I deduce (am I wrong?) that there are many ...
- 111
6
votes
0
answers
172
views
Does there exist a plane curve such that it has the heart curve as catacaustic?
Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$.
The catacaustic ...
- 565
0
votes
0
answers
30
views
Application of greedy approach for optimization
I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$
where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 ...
- 23
5
votes
0
answers
137
views
Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products
Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
- 351
1
vote
2
answers
276
views
Does $\mathbb Z^n$ contain $A_n$?
Are there any positive integer $n > 3$ such that the root lattice $A_n$ is contained in $\mathbb Z^n$?
- 646