Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.
2
votes
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
$$n_k:=\#\{g\in G: |g|...
19
votes
1answer
521 views
Diameter of a quotient of the infinite dimensional sphere
Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.
Assume that the action $...
4
votes
1answer
76 views
Clustering distance
Is there a good notion of distance between partitions of a (fixed, finite) set? The context is this: suppose I have a clustering algorithm, which clusters points using some method or other. Now, I ...
12
votes
2answers
305 views
Geodesic current supported on a pencil?
Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...
11
votes
0answers
149 views
Elkies points in the plane of a triangle $ABC$
Noam Elkies proved that if $x,y,z$ are positive numbers, then there is a unique point $P$ inside $ABC$ such that the inradii $r_a,r_b,r_c$ of the triangles $BPC, CPA, APB,$ respectively, satisfy
$$ ...
14
votes
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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 $$\forall x \in M, \...
3
votes
2answers
208 views
Minimum weight triangulation of lattice points in a circle
Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points
$S$ inside or on the circle $C$ of radius $r$ centered on the origin.
Let $P$ be the convex hull of $S$; so $P$ is inscribed ...
1
vote
1answer
127 views
Hausdorff distance is a lower (or upper bound) for what probability metric?
In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that
$$
d(A, B) \le W(\mu|_A, \mu|_B),
$$
where
$d(A, B):= \...
2
votes
0answers
37 views
Dimensions and volumes of continuous images of closed Riemannian manifolds
Suppose $M$ is an $m$-dimensional closed Riemannian manifold and $f \colon M \to \mathbb{R}^{n}$ is continuous. I'm interested in the case when $M=\mathbb{T}^{m}$. Let $\mathcal{M}_{f} := f(M)$. Let $...
1
vote
1answer
59 views
smallest square containing k non-overlapping equal rectangles at any orientation
This seems like something that should have a known answer, but I haven't found it after some time alternating between searching and generating multiple pages of algebra. I'm interested in $k=4$ and $...
0
votes
1answer
77 views
Sampling a uniformly distributed point INSIDE a hypersphere?
There is a simple algorithm to pick a random point ON an $n$-dimensional hypersphere.
Is there one to sample a point from inside it? (Sampling points from a hypercube and rejecting them if they are ...
4
votes
1answer
105 views
Energy in doubling measure metric spaces
Let $(X,\mu, d)$ be a metric measure space where $\mu$ is a doubling measure. For a relatively compact set $U\in X$ consider the following quantity
$$I(U,\mu,d)=\int_U \int_U \log^2(d(x,y)) d\mu(x) d\...
4
votes
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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 ...
