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

**2**

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99 views

### What is the name of the two points in this triangle construction? Or are the points known?

Let $ABC$ be a triangle and $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively.
From my construction by GeoGebra, I found two special points as ...

**1**

vote

**0**answers

55 views

### DIstance on a Riemannian manifold

Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example:
...

**6**

votes

**0**answers

72 views

### Geometric mean of three or more positive definite matrices

The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...

**-2**

votes

**0**answers

36 views

### Plotting a 2D circled based on a 3D sphere? [closed]

I'm currently working on a project where I have to - using a computer and a screen - draw a sphere that rotates. Due to limited computing power, I have to do this two-dimensionally.
This is ...

**4**

votes

**0**answers

127 views

### Uniqueness of the boundary of a hierarchically hyperbolic group

Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...

**11**

votes

**0**answers

187 views

### Why are the medians of a triangle concurrent? In absolute geometry

This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...

**0**

votes

**0**answers

52 views

### Which $CAT(0)$-polygonal complexes are median spaces?

$CAT(0)$-polygonal complexes are simply connected collections of glued (on their faces) polyhedra of varying dimension such that each link is flag.
Which $CAT(0)$-polygonal complexes with appropriate ...

**3**

votes

**1**answer

119 views

### Are $CAT(0)$-polygonal complexes median spaces?

A median space is a metric space $X$ for which for any three points $x, y , z \in X $ there exists a unique point $m$ such that $d(x,m)+ d(m, y)= d(x , y ), d(x,m)+ d(m, z)= d(x , z ), d(y,m)+ d(m, z)=...

**1**

vote

**1**answer

122 views

### Approximate the following series on the euclidean grid

I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the ...

**2**

votes

**1**answer

89 views

### Why the VC dimension of triangles in 2D space is not greater than 7?

I understand that there are sets of 7 points on a circle that can be fully
shattered using triangles.But, it is not clear to me why it cannot shatter 8 points.
Is there any intuitive way of arriving ...

**13**

votes

**1**answer

261 views

### Can the graph of a symmetric polytope have more symmetries than the polytope itself?

I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...

**9**

votes

**2**answers

1k views

### Do two new special points in any triangle exist?

There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.
Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $...

**2**

votes

**0**answers

75 views

### Lebesgue density theorem for “doubling uniformly covering collections of subsets”

I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically
Let $(...

**8**

votes

**1**answer

204 views

### Intersection of nested open ball in complete metric spaces is nonempty?

My question is that whether the following statement is true or not.
In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x_i, r_i)\}_{i=1}^\infty$ satisfies
$$
\exists \epsilon > ...

**3**

votes

**1**answer

54 views

### Maximal edge length of symmetric polytopes

For me, a polytope is the convex hull of finitely many points. It is said to be vertex-transitive / edge-transitive if its symmetry group acts transitively on its vertices / edges. Let's call a ...

**2**

votes

**0**answers

110 views

### A problem on real analysis related to Hilbert's fourth problem

This is an extensive re-write of a question I deleted and which had basically the same title.
Identify the cylinder $S^1 \times \mathbb{R}$ with the space of (co)oriented lines in the plane by ...

**2**

votes

**1**answer

238 views

### About the growth rate of a group

Let $G$ be a f.g. group and $d$ be a word metric w.r.t. a symmetric generating set. For $g\in G$, define $|g|:=d(g,e)$, where $e$ is the group identity. For $k\in\mathbb N$, put
$$n_k:=\#\{g\in G: |g|...

**19**

votes

**1**answer

521 views

### Diameter of a quotient of the infinite dimensional sphere

Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.
Assume that the action $...

**4**

votes

**1**answer

76 views

### Clustering distance

Is there a good notion of distance between partitions of a (fixed, finite) set? The context is this: suppose I have a clustering algorithm, which clusters points using some method or other. Now, I ...

**12**

votes

**2**answers

305 views

### Geodesic current supported on a pencil?

Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...

**11**

votes

**0**answers

149 views

### Elkies points in the plane of a triangle $ABC$

Noam Elkies proved that if $x,y,z$ are positive numbers, then there is a unique point $P$ inside $ABC$ such that the inradii $r_a,r_b,r_c$ of the triangles $BPC, CPA, APB,$ respectively, satisfy
$$ ...

**14**

votes

**2**answers

384 views

### The Disco Ball Problem

Let me first give some of a background as to where I got this problem. I had a math teacher ask me a few months ago: "How many 1 unit by 1 unit squares could one fit on a sphere with a radius of 32 ...

**5**

votes

**0**answers

60 views

### Hölder isoperimetric problem

Denote by $S_r$ the usual circle of radius $r$, with the path metric ($d(x,y) = r\theta$, where $\theta$ is the angle between the vectors $x$ and $y$), and let $\alpha \in (1/2,1)$. Consider the ...

**1**

vote

**0**answers

85 views

### Checking Planar Convexity of 4 Points with Stewart's Formula

Is the following conjecture correct?
Conjecture:
If $A,B,C,D$ are four points in general position in the euclidean plane, with
$a:=\|C-B\|,\ \ b:=\|C-A\|,\ \ c:=\|B-A\|$
$a':=\|D-A\|,\...

**2**

votes

**0**answers

52 views

### 8-partition of a planar convex body by 4 concurrent lines

It is known1 that any convex body $K$ in the plane can be
partitioned into $6$ equal-area pieces by $3$ concurrent lines
which meet at a point in $K$.
Call this a $6$-partition.
This result cannot be ...

**1**

vote

**1**answer

145 views

### A possible characterization of the cube?

Let $P$ be the $1$-skeleton of a convex polyhedron fixed in $\mathbb{R}^3$,
and $|P|$ the sum of the Euclidean lengths of the edges of $P$.
Let $P_1, P_2, P_3$ be the perpendicular projections of $P$
...

**5**

votes

**1**answer

212 views

### A possible characterization of sphere or projective space

Is there a compact Riemanian manifold $M$ not diffeomorphic to sphere or real or complex or quaternion projective space which admit a diffeomorphism $f$ with the property that $$\forall x \in M, \...

**3**

votes

**2**answers

208 views

### Minimum weight triangulation of lattice points in a circle

Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points
$S$ inside or on the circle $C$ of radius $r$ centered on the origin.
Let $P$ be the convex hull of $S$; so $P$ is inscribed ...

**1**

vote

**1**answer

127 views

### Hausdorff distance is a lower (or upper bound) for what probability metric?

In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that
$$
d(A, B) \le W(\mu|_A, \mu|_B),
$$
where
$d(A, B):= \...

**2**

votes

**0**answers

37 views

### Dimensions and volumes of continuous images of closed Riemannian manifolds

Suppose $M$ is an $m$-dimensional closed Riemannian manifold and $f \colon M \to \mathbb{R}^{n}$ is continuous. I'm interested in the case when $M=\mathbb{T}^{m}$. Let $\mathcal{M}_{f} := f(M)$. Let $...

**1**

vote

**1**answer

59 views

### smallest square containing k non-overlapping equal rectangles at any orientation

This seems like something that should have a known answer, but I haven't found it after some time alternating between searching and generating multiple pages of algebra. I'm interested in $k=4$ and $...

**0**

votes

**1**answer

77 views

### Sampling a uniformly distributed point INSIDE a hypersphere?

There is a simple algorithm to pick a random point ON an $n$-dimensional hypersphere.
Is there one to sample a point from inside it? (Sampling points from a hypercube and rejecting them if they are ...

**4**

votes

**1**answer

105 views

### Energy in doubling measure metric spaces

Let $(X,\mu, d)$ be a metric measure space where $\mu$ is a doubling measure. For a relatively compact set $U\in X$ consider the following quantity
$$I(U,\mu,d)=\int_U \int_U \log^2(d(x,y)) d\mu(x) d\...

**4**

votes

**0**answers

70 views

### Geometric meaning of the chi-square “measure of association”

In Statistics, there's a standard test of independence of two random variables taking values in finite sets $I,J$. It relies on the computation of $\chi$-square statistics,
$$
\chi^2:=\sum_{(i,j)\in ...

**3**

votes

**2**answers

105 views

### Cone-Torus intersection in 3D

Problem. I have a solid torus and a solid cone in $\mathbb R^3$ and need an efficient algorithm that determines if they intersect or not.
The center of the torus is at a given position $\mathbf p \in ...

**7**

votes

**1**answer

128 views

### existence of riemannian metric on $\text{SL}_3(\mathbb{R})$ with special geodesics

Is there a left-invariant Riemannian metric on $\text{SL}_3(\mathbb{R})$ for which the geodesics (with respect to the corresponding Levi-Civita connection) through the identity are exactly the ...

**7**

votes

**1**answer

253 views

### Axioms of length

Assume I want to define length of plane curves axiomatically.
It seems to be reasonable to assume that
The length of a unit segment is 1;
Congruent curves have equal lengths;
Length is additive with ...

**28**

votes

**1**answer

567 views

### Gromov-Hausdorff distance between a disk and a circle

The Hausdorff distance between the closed unit disk $D^2$ of $\mathbb R^2$ (equipped with the standard Euclidean distance) and its boundary circle $S^1$ is obviously one.
Interestingly, the Gromov-...

**0**

votes

**0**answers

79 views

### Bisecting plane convex sets wrong intuition

Let $S$ be a compact, convex set in $\mathbb{R}^2$ with non-empty interior and let $p$ be the center of mass of $S$. I was wondering if any line through $p$ cuts $S$ in two sets with equal area.
My ...

**1**

vote

**1**answer

163 views

### Group action on quasi-isometric geodesic metric space [closed]

If a group $G$ acts on a geodesic metric space $X$, then does $G$ act on a geodesic metric space $Y$ which is quasi-isometric to $X$?

**5**

votes

**0**answers

173 views

### Which subspaces of $\ell_p^n$ are isometric?

This question is similar to the one asked here:
Extending linear isometries from subspaces of $\ell_p^n$
Let $p$ be an even integer. Let $X,Y$ be subspaces of $\ell_p^n$, and let $U : X \to Y$ be a ...

**0**

votes

**0**answers

29 views

### General conditions for Gaussian isoperimetric inequaliteis

Context
I'm doing some work in which i need to show that the blow-up $B_\epsilon$ of a Borel set $B$ has large measure for $\epsilon$ sufficiently large. I've found a very attractive tool for doing ...

**5**

votes

**1**answer

103 views

### When can the metric be reconstructed (up to scaling) from knowing the conjugate points?

Let $M$ be a smooth manifold of dimension $\geq 2$. Let $g$ be a complete Riemannian metric on $M$. Let $C \subseteq M \times M$ be the set of pairs of $g$-conjugate points.
The set $C$ doesn't ...

**2**

votes

**2**answers

79 views

### Monotonicity for the side lengths of stars inscribed in regular polygons

Fix integers $l\ge 1$ and $n \ge 3$, and let $P_n$ denote the boundary of the regular $n$-sided polygon in the plane. We define a $(2l+1)$-pointed equilateral star to be a cyclically ordered list of ...

**0**

votes

**0**answers

58 views

### Applying a piecewise linear function to vertices of a polytope while remaining in facet representation

Let $P \subseteq \mathbb{R}^d$ be a polytope with vertices $V$, and let $f : \mathbb{R}^d \to \mathbb{R}$ be a function. Let $P' \subseteq \mathbb{R}^{d+1}$ be the polytope with vertices $\{(v, f(v)) \...

**2**

votes

**0**answers

74 views

### Totally geodesic submanifold of codimension 1 in noncompact Riemannian manifold

Assume that $M$ is a noncompact complete simply connected manifold of nonnegative sectional curvature. Then by Soul theorem, it has a soul $S$.
Question 1 : Fix a point $p\in S$. Then there is a ...

**2**

votes

**2**answers

224 views

### Totally geodesic submanifold of codimension 1

This question is inspired by question in reference.
Question : If $M$ is a simply connected closed Riemannian manifold of nonnegative sectional curvature, then there is a totally geodesic ...

**3**

votes

**0**answers

74 views

### Is each metric continuum $\ell_p$-chain connected?

This problem was motivated by the MO problems:
"Running most of the time in a connected set", "Is every metric continuum almost path connected?" and "Are $\varepsilon$-connected components dense?".
...

**1**

vote

**0**answers

55 views

### Can sufficiently symmetric polytopes be uniquely reconstructed from their 1-skeleton?

General convex polytopes can not be uniquely reconstructed from their 1-skeleton1, as explained here. Not even the dimension is known from the skeleton, as e.g. the complete graph $K_n,n\ge 5$ is the ...

**6**

votes

**1**answer

113 views

### Are $\varepsilon$-connected components dense?

Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of ...