Questions tagged [mg.metric-geometry]

Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.

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3 votes
1 answer
152 views

Distance function and geometry of the set

Let $X \subseteq \mathbb{R}^n$ be a closed $d$-dimensional regular set (i.e. for any $x \in X$ and $0<r< \text{diam(X)}>$, $\mathscr{H}^d(B(x,r)) \sim r^d$ ) which has the property that for ...
0 votes
0 answers
47 views

Random sequential adsorption of rectangles with random aspect ratio

Jian-Sheng Wang, in his 1994 article A fast algorithm for random sequential adsorption of discs (arXiv link), considered deposition only on (small) squares which are not fully covered by the excluded ...
1 vote
1 answer
1k views

Closed-form upper-bounds for Wasserstein distance between finite measures

Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
6 votes
1 answer
210 views

Is Sydler's theorem concerning Dehn invariants constructive?

Sydler proved something of a converse to Dehn's negative resolution of Hilbert's 3rd problem. To quote Wikipedia, Sydler showed that "every two Euclidean polyhedra with the same volumes and Dehn ...
7 votes
0 answers
446 views

Applications of the co-area formula

Kirchheim [2] generalized the classical area formula to the case of Lipschitz mappings into metric spaces. Ths paper is well known and widely cited. The area formula is a special case of the co-area ...
18 votes
1 answer
1k views

The group of isometries of a manifold is a Lie group, isn't it?

Let $M$ be a connected finite dimensional topological manifold and $g$ be any metric on it that induces the topology of $M$ ($g$ is not a Riemannian metric). How to prove that the group of isometries ...
1 vote
1 answer
61 views

How to verify that an element in the root lattice is an imaginary root of a non-hyperbolic root system?

In my research I encounter some elements in a root lattice and I would like to verify that these elements are imaginary roots. Consider the root system $J_{6, 11}$ with the following Dynkin diagram: \...
2 votes
0 answers
131 views

Optimal way to group points in the plane into clusters

Consider a strictly decreasing sequence $d = (d_k)_{k\ge 1}$ of distances in $(0,1)$. Given a constant $C>2$, we say that $d$ has the $C$-grouping property if any finite non-empty subset $S$ (of ...
0 votes
0 answers
114 views

Are there any special relations among the 27 lines on a cubic surface?

If we consider 3 skew lines and the hyperboloid formed by them, any other line on the hyperboloid intersects these three lines. Moreover, the lines orthogonal to the plane containing a point on the ...
6 votes
2 answers
518 views

Which groups are doubling?

A metric space $(M,d)$ is doubling if there exists $n$ such that every ball of radius $r$ can be covered by $n$ balls of radius $r/2$, for all $r$. For which f.g. groups $G$ and finite symmetric ...
1 vote
1 answer
322 views

Exact volume calculation of a polytope is NP hard under which restrictions?

Computing the exact volume of a polytope given in half space representation seems to be NP-hard. One paper I found proved it is hard for rational coefficients. (However, the paper itself was behind a ...
7 votes
1 answer
297 views

Assigning a "canonical geometry" to a Seifert surface

I originally posted this on stackexchange, but it hasn't gotten an answer. I hope it's not inappropriate for this forum. Suppose I have a knot $K: S^1 \hookrightarrow S^3$ with minimal genus Seifert ...
8 votes
1 answer
998 views

Possible new theorem in plane geometry encompassing 5 famous geometry theorems

I am looking for a proof of a generalization Napoleon theorem, Bottema theorem and Brahmagupta theorem and van Aubel theorem, and Finsler–Hadwiger theorem in one configuration, as follows: Let four ...
1 vote
0 answers
81 views

Constructive way to optimally cover a compact subset of Euclidean space

Let, $(X,d)$ be a simply connected compact subset of $\mathbb{R}^d$ with non-empty interiorn, let $d$ denote the Euclidean metric, and let $\varepsilon>0$. Is there a way to iteratively select ...
4 votes
1 answer
199 views

Mass distributions for high dimensional simplex and cross polytope

In this question, it is shown that the radial mass distribution of an $n$-cube (i.e. the probability density for the distance from a point selected uniformly from within an $n$-cube to the cube's ...
5 votes
1 answer
324 views

Discovered 240 new circles associated with Pascal's line

I am looking for a proof or a reference request for a problem as follows: Problem: Let a cyclic hexagon with sidelines $l_1$, $l_2$, $l_3$, $l_4$, $l_5$, $l_6$ and $l_1 \cap l_4 =A$, $l_3 \cap l_6 = ...
14 votes
1 answer
236 views

Must a path of compact sets in $X$ descend to a path in $X$?

(I am most interested in the case $X=\mathbb R^2$, but of course one could ask the same question for manifolds, or metric spaces in general.) Let $\text{Com}(\mathbb R^2)$ denote the space of nonempty ...
3 votes
1 answer
133 views

Tangencies of Villarceau circles in a 3D Steiner chain

Consider a Steiner chain made of an arbitrary number $n$ ($\geq 3$) of spheres (not circles, spheres), as in the picture below with $n=6$ (so it is a so-called Soddy hexlet). I've found this picture ...
5 votes
1 answer
173 views

Finding a superbase in a lattice of Voronoi first kind

An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that $\{b_1,\ldots,b_n \}$ is a ...
3 votes
1 answer
175 views

On some centers of convex regions based on partitions

These questions are inspired by Yaglom and Boltyanskii's 'Convex figures'. Winternitz Theorem: If a 2D convex figure is divided into 2 parts by a line $l$ that passes through its center of gravity, ...
51 votes
4 answers
6k views

what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics

In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of ...
4 votes
0 answers
197 views

How to find the dimension of the polar cone of a convex cone generated by some given vectors

Suppose we have access to a generating set $\{v_1, ..., v_k\}\subseteq\mathbb{R}^n$ of the convex cone $C=cone(v_1, ..., v_k)$, where $cone(\cdot)$ is the conical hull (i.e. nonnegative span) of ...
10 votes
1 answer
290 views

Vietoris-Rips complex and coarse geometry

Let $K$ be an infinite countable subset of Euclidean space $E$ such any point of $E$ is within distance 1 of some point of $K$. In the language of John Roe's "coarse geometry", this implies that $K$ ...
4 votes
1 answer
199 views

$\varepsilon$-net of a $d$-dimensional unit ball formed by power set of $V = \{+1, 0 -1\}^d$

I have a set of $d$-dimensional vectors $V = \{+1, 0, -1\}^d $. Then $P(V)$ constitutes the power set of $V$. I now construct a set of unit vectors $V_{\mathrm{sum}}$ from the power set $P(V)$ such ...
0 votes
0 answers
47 views

Background smooth geometry vs background edge geometry

In this note, page 8 and page 9, the Kähler–Einstein edge equation is written first in (3.7) in terms of a model edge metric and in (3.11) in terms of a smooth Kähler metric. What is the distinction ...
8 votes
1 answer
449 views

Convergence in the Caratheodory sense and Hausdorff sense

Among Jordan domains, I understand that Caratheodory convergence is weaker than Hausdorff convergence. But if a sequence of Jordan domains all have rectifiable boundary whose arc length are all $L$, ...
4 votes
0 answers
213 views

Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$

Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric. I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
2 votes
1 answer
272 views

How to fold a tesseract from L-unfolding? [closed]

I came across an image, that show really simple unfold of 4-dim cube. https://arxiv.org/pdf/1512.02086.pdf here at #2.1, and 120 here https://page.mi.fu-berlin.de/moritz/mo/198722/unfoldings.html. ...
4 votes
3 answers
319 views

Minimal data required to determine a convex polytope

Let $P\subset \Bbb R^d$ be a convex polytope. Suppose that I know its combinatorial type (aka. the face-lattice), the length $\ell_i$ of each edge, and the distance $r_i$ of each vertex from the ...
2 votes
1 answer
705 views

Vafa's semi-Ricci flat metric

Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and noncompact Calabi-Yau manifolds. Nuclear Physics B,...
5 votes
0 answers
88 views

Which polytopes can be deformed while keeping their edge-lengths?

Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while keeping its combinatorial type, and keeping its ...
1 vote
1 answer
115 views

Property of convex polygons on integer lattice structures

Another graduate student and I are working on an research project and are looking for a paper or other source that has a proof for a result about polygons on an integer lattice structure. Suppose you ...
1 vote
0 answers
85 views

Recursive expression of Lebesgue measure for balls with removed intersection

This is not the most theoretically challenging question; rather it is more of a reference request for a simple formula (which must be known). Let $\left\{B_{\epsilon_n}(x_n)\right\}_{n=0}^N$ be a set ...
2 votes
1 answer
478 views

A geometric approach to the odd perfect number problem?

Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$. Let $h(n) = J_2(n)$ be the second Jordan totient function. Define: $$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
1 vote
0 answers
46 views

Bound for the cardinality of maximal $r$-separable subsets contained in a ball of radius $R$ in $\mathbb R^d$

Let $B$ be a closed ball in $\mathbb R^d$ of radius $R$ and let $N=N_R(r)$ denote the maximal cardinality of the $r$-separated sets (meaning any two points in this set have distance at least $r$) that ...
2 votes
1 answer
364 views

Gromov-Hausdorff distance between weighted tree graphs

I would like to measure the similarity between a pair of weighted tree graphs. According to this post, this can be done by regarding the trees as metric spaces and then applying the Gromov-Hausdorff ...
1 vote
0 answers
106 views

Is the mean center of the vertices of a convex polygon always inside the polygon? [closed]

As simple as that. I'm doing an R program where I need to order clockwise a bunch of points that describe a regular polygon and to do that I figured I could find a point inside, change to polar ...
2 votes
1 answer
160 views

Can we define geodesic in the space of compactly supported functions?

From Wikepedia, the definition of geodesic is stated as: A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...
3 votes
1 answer
103 views

Solid angles at points in an orthosimplex

Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ ...
19 votes
1 answer
873 views

Can every simple polytope be inscribed in a sphere?

It is known that not every convex polytope (even polyhedron, e.g. this one) can be made inscribed, that is, we cannot always move its vertices so that all vertices end up on a common sphere, and the ...
2 votes
2 answers
465 views

Reference question: Poncelet theorem

A very famous theorem of Poncelet states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in ...
2 votes
0 answers
141 views

What practically computable homotopy and/or (co)homology theories are known for finite (di)graphs, metric spaces, etc?

Of late I have taken to applying Dowker homology and the path homology theory of Grigor'yan et al. like a hammer to various relations and/or digraphs that have looked like nails. At the same time, I ...
7 votes
0 answers
296 views

The space of $p$-adic norms

The 1963 paper by Goldman and Iwahori The space of $p$-adic norms deals with the space of norms on a finite dimensional vector space $E$ over a locally compact complete discrete valuation field $K$. I ...
1 vote
0 answers
275 views

Minkowski (box-counting) dimension of generalized Cantor set

I'm trying to solve this problem. For $0<\alpha, \beta<1,$ let $K_{\alpha, \beta}$ be the Cantor set obtained as an intersection of the following nested compact sets. $K_{\alpha, \beta}^{0}=[0,...
5 votes
2 answers
189 views

Fast algorithms for calculating the distance between measures on finite ultrametric spaces

Let $X$ be a finite ultrametric space and $P(X)$ be the space of probability measures on $X$ endowed with the Wasserstein-Kantorovich-Rubinstein metric (briefly WKR-metric) defined by the formula $$\...
0 votes
1 answer
283 views

Dither in Leech lattice quantization!

Can you please help me how to generate a dither signal $\mathbf{U}$, where $\mathbf{U}$ is a random vector of length 24 that is uniformly distributed over the Voronoi region of the Leech lattice. Best,...
0 votes
1 answer
91 views

What is sequential boundary of a $\delta$-hyperbolic space and how is the Gromov product extended to the boundary?

I have been reading up on $\delta$-hyperbolic spaces. But I am not getting a clear idea of sequential boundary of $\delta$-hyperbolic spaces and how the Gromov product is extended to it. Could ...
37 votes
3 answers
3k views

What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...
17 votes
4 answers
1k views

Can I build infinitely many polytopes from only finitely many prescribed facets?

Given a finite set of convex $d$-dimensional polytopes $\mathcal P$, for some $d\ge 2$. Question: Is it true that there are only finitely many different convex $(d+1)$-dimensional polytopes whose ...
5 votes
1 answer
112 views

Packing in uniform domains

Given $N$ points $X:=(x_i)_{i \in \{1,..,N\}}$, we now define a score function $S:X \rightarrow \mathbb{N}$ that is $S(X)= \sum_{i=1}^N S(x_i)$ where the score of $S(x_i)$ is $$S(x_i) = 2* \vert \{x_j;...

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