# Questions tagged [mg.metric-geometry]

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

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### Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?

Per the title, what are some of the oldest (non-analytic) geometry books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there.

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### How to compute the parameters of circumscribed hypershpere? [on hold]

Assume I have an $n$-dimensional simplex on the points $x_0, ..., x_n$ where each $x_i \in \mathbb{R}^n$. I would like to obtain the parameters (center and radius) of it's circumscribed n-dimensional ...

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### Metrically Ramsey ultrafilters

On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the ...

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### Bishop-Gromov inequality strengthened for anisotropic metrics?

The Bishop-Gromov inequality provides an upper-bound on the rate of growth of volume of a ball of radius $r$ in spaces that have a lower-bound on the Ricci curvature, $Ric \geq (n-1)K$.
(I am ...

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### Polygons such that $n^2 $ times magnification of a polygon could be covered by exactly $n^2$ original polygon

While studying about covering problems in combinatorics, I got to a simple question:
What polygons can be covered exactly, without any area that is covered twice or area that is outside the covered ...

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### Sums of squared distances between points on an $n$-sphere

I have discovered the following results about the sums of squared distances between points on an $n$-sphere (and proved them). To the best of my knowledge (and my advisor's knowledge), these results ...

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### Uniform versus non-uniform group stability

Group stability considers the question of whether "almost"-homomorphisms are "close to" true homomorphisms. Here, "almost" and "close to" are made rigorous using a group metric.
More precisely, ...

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### Metric 1-current decomposition

I've been reading Paolini-Stepanov arcticle and in section 4, at page 6, they define a metric current from a transport:
$$T_{\eta}(\omega)=\int_{\Theta}[[\theta]](\omega)d\eta(\theta),$$
which ...

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### Uniformly Converging Metrization of Uniform Structure

This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...

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### Is a polytope with vertices on a sphere and all edges of same length already rigid?

Let's say $P\subset\Bbb R^d$ is some convex polytope with the following two properties:
all vertices are on a common sphere.
all edges are of the same length.
I suspect that such a polytope is ...

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### $L^{2}$ Betti number

Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...

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### Which cubic graphs can be orthogonally embedded in $\mathbb R^3$?

By an orthogonal embedding of a finite simple graph I mean an embedding in $\mathbb R^3$ such that each edge is parallel to one of the three axis. To avoid trivialities, let's require that (the ...

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### Criterion for visuality of hyperbolic spaces

I am trying to understand the following sentence on p. 156 of Buyalo-Schroeder, Elements of asymptotic geometry: "Every cobounded, hyperbolic, proper, geodesic space is certainly visual."
Let $X$ be ...

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### An isoperimetric inequality for curve in the plane?

Let $f(x,y)=0$ be a (smooth) simple closed curve $C$ on the plane and $R$ the region bounded by $C$ (appropriately oriented). Assume the origin lies in the interior of $R$.
QUESTION. Let $r=\sqrt{x^...

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### What is the meaning of Conjugate radius and Injectivity radius?

I review the text book of differential geometry and I find that the conjugate radius and injectivity radius are still enigmatic for me. Here is a quetion which I confuse it. I don't think it is true, ...

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### Valid metric on a hyperbolic space

Note: originally posted on math.SE.
I'm looking at the distance that's defined in this paper on Poincaré Embeddings:
$d(\mathbf{u}, \mathbf{v}) = \operatorname{arccosh} \left(1 + 2\frac{\left\| \...

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### Can 4-space be partitioned into Klein bottles?

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles,
or into disjoint unit circles, or into congruent copies of a real-analytic curve
(Is it possible to partition $\mathbb R^3$ ...

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### A questions concerning Laguerre/Voronoi tessellations

Fix $n>1$ distinguished points $x_1,\ldots, x_n\in \mathbb R^d$, the Voronoi tessellations are the subsets $V_1,\ldots V_n\subset\mathbb R^d$ defined by
$$V_k~~ := ~~ \big\{x\in\mathbb R^d:\quad |...

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### Estimation of the Gromov–Wasserstein distance of spheres

Let $(X,d_X,\mu_X)$ and $(Y,d_Y,\mu_Y)$ be two metric measure spaces. A probability measure $\mu$ over $X\times Y$ is called a coupling if $(\pi_1)_\sharp \mu=\mu_X$ and $(\pi_2)_\sharp \mu=\mu_Y$. We ...

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### Equal volume and projections

Given three unit vectors $u_1,u_2,u_3$ in $\mathbb{R}^3$, can we find some body $K \subset \mathbb{R}^3$ (probably convex) such that the following three things hold
(1) $|P_{u_1^\perp}K|=|P_{u_2^\...

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61 views

### Probability of two Points being divided by an high-Dimensional Hyperplane

I have two points $x_1,x_2 \in \mathbb S^n $ which are distant $d$ from each other, where $d<<1$.
I also have a vector $v$ sampled uniformly at random from $\mathbb S^n$.
What is the ...

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### Locally compact metric spaces whose group of isometries generated by isometries of small displacement is transitive enough

I wasn't sure how to make the title any more precise than that.
Let $(X,d)$ be a locally compact metric space. For any $\varepsilon>0$ let $\mathrm{Aut}_\varepsilon (X)$ be the subgroup of the ...

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### Modulus of image of a curve family in a rectangle

I don't expect to get a positive answer to this question but I may as well try.
Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...

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### A special connected subset of the Cantor fan

Is there a dense connected subset $X$ of the Cantor fan $$(C\times [0,1])/(C\times \{1\})$$ such that for every two connected subsets $X_1,X_2\subseteq X$, the intersection $X_1\cap X_2$ is connected?
...

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### Reference request: $\alpha$-Hölder spaces as double duals

If $(X,d)$ is a complete metric space, we define the $\alpha$-Hölder class $\Lambda_\alpha(X)$ as the subset of $C_b(X)$ satisfying that
$$
\sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}.
$$
...

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**1**answer

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### parametrize triangles meeting certain conditions

Consider triangles with angles alpha, beta, gamma such that gamma = 2 alpha, and sides (a,b,c) are integers. I want to parametrize such triangles by a single integer or rational variable.

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### Homeomorphism type of the horofunction boundary for nilpotent Lie groups

Consider a metric space $(X,d)$ and fix a base point $w$. A horofunction is a function of the form
$$\beta_y(x)=d(x,y)-d(w,y).$$
The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-...

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### Geometry of complements to compacts of codimension 2

Let $K\subset \mathbb{R}^n$ be a (nonempty) compact of covering dimension $\le n-2$. In particular, $K$ does not separate $\mathbb{R}^n$ (even locally). I will equip $M=\mathbb{R}^n-K$ with the ...

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### Suppose that a metric space allows David–Semmes regular map to some Euclidean space. Does it allow bi-Lipschitz embedding into some Euclidean space?

I want to ask about the progress on Question 8 from "Thirty-three yes or no questions about mappings, measures, and metrics" by Juha Heinonen and Stephen Semmes. Is it still open? If yes, was some ...

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### Non-homeomorphic computable metric spaces whose computable points are computably homeomorphic

This is a follow-up of sorts to an earlier question on mine in that it should be easier to construct a positive example of this, if it exists.
To be clear about definitions, a computable metric space ...

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### Convexity in co-ordinate charts of geodesic balls

Let $g_{ij}$ be a Riemannian metric tensor on an open subset $U\subseteq \mathbb{R}^n$, and let $p\in U$.
I would guess the following is true:
for $\epsilon$ sufficiently small, the $g$-geodesic ...

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### Extremal metric for image of a curve family

Let $U\subset \mathbb{C}$ be a domain and $\Gamma$ some family of curves in $U$ with $\textrm{mod}(\Gamma)<\infty$ and such that $\rho$ is an extremal metric for the modulus. Suppose we are given a ...

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### Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$

An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane.
The following questions are motivated by Anton Petrunin's Disc bounded ...

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### Simple connectedness under a metric undistortion condition: on a tricky point in an argument of Gromov

The context
I have been reading Gromov's Metric Structures..., and came upon result 1.14.(a), page 11, which states the following.
Let $K\subset\mathbb R^d$ be a compact subset, and $d_\ell$ its ...

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### Explicit Quasisymmetric embedding into Euclidean space

It is known that every doubling metric space admits quasisymmetric map into Euclidean space. My question is, is there a known explicit (closed-form) quasisymmetry from the Heisenberg group into a ...

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### Is a vertex- and edge-transitive polytope already a uniform polytope?

I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive).
Question: Is every such ...

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### How to construct a nice homotopy?

Let $(M,g)$ be a closed simply-connected Riemannian manifold. Can we find a constant $C$, which depends on $M$, such that for any closed curve $\alpha_0$ with $L(\alpha_0) \le 1$, there exists a ...

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### Is there any work in topological data analysis on something like “Voronoi complexes”?

Given a finite set $X \subset \mathbb{R}^n$, we can of course construct the corresponding Čech or Vietoris-Rips filtration. At each level of this filtration the scale parameter is fixed and unrelated ...

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### Lower bound to $\epsilon$-expansion of a subset of a half-sphere

Below are two known lemmas on a $d$-dimensional sphere (related to the isoperimetric inequality). I would like to know: does a similar statement like this holds for a $d$-dimensional dome also (i.e. ...

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### Metric Measure Space: A Basic Question

I understand the basic definition of a metric measure space to be the following:
A metric measure space is a triple of a space $X$, metric $d$, and measure $m$: $(X,d,m)$ in the sense that the ...

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### Can all countable $CAT(0)$ cube complexes be isometrically embedded in $l^1(\mathbb{N},\mathbb{R})$?

In this paper (theorem 2), Chepoi & Hagen say
There exists an infinite $CAT(0)$ cube complex $X$ with constant maximum degree which cannot be isometrically embedded into a Cartesian product of ...

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### Algorithm for MaxMin diversity problem on hypercube?

The Maximum Minimum Diversity Problem (MMDP) can be roughly stated as follows:
Let $S$ be a set of points, $k \in \mathbb{N}$. Find the $k$-element subset $M$ of $S$ such that the minimal distance $...

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### What kind of set is this, spanned by two positive definite matrices?

Let $A$ and $B$ be Hermitian positive definite $n\times n$ matrices over $\mathbb C$ or $\mathbb R$. Then for real $k,\ell,$ the matrix $A^kB^\ell A^k$ is well-defined and again Hermitian positive ...

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### Sum of squared nearest-neighbor distances between points on the sides of a rectangle

For positive real numbers $a,b$, let $R$ denote the $a\times b$ rectangle $[0,a]\times[0,b]$. Let $A_1,\dots,A_4$ be points on the sides of $R$, one point on each side. For each $j=1,\dots,4$, let $...

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### Is the metric completion of a Riemannian manifold always a geodesic space?

A length space is a metric space $X$, where the distance between two points is the infimum of the lengths of curves joining them. The length of a curve $c: [0,1] \rightarrow X$ is the sup of
$$ d(c(0),...

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### Sum of squared nearest-neighbor distances between points in a square

Let $\square_2=\{(x,y): 0\leq x, y\leq1\}$ be the unit square in $\mathbb{R}^2$. Take $n>1$ points $P_1, \dots, P_n\in\square_2$.
Denote the distances $d_j=\min\{\Vert P_k-P_j\Vert: k\neq j\}$, ...

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### Geodesics on the sphere

In a few days I will be giving a talk to (smart) high-school students on a topic which includes a brief overview on the notions of curvature and of gedesic lines. As an example, I will discuss flight ...

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### The typical amount of lattice points in a set as dimension goes off to infinity

I have encountered a geometry-of-numbers-type problem when calculating an entropy in a lattice communications scheme:
$L^{(n)}$ is a lattice uniform over all those in $\mathbb{R}^n$ with base ...

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### Can bellows make loops?

Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?

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### non-proper parabolic isometries of hyperbolic spaces

In his seminal paper on hyperbolic groups (see Section 8.1) Gromov defines an isometry $f$ of a hyperbolic space $X$ to be parabolic if the orbit of any point $x\in X$ under the action of $\langle f\...