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

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Starlike curve tangent condition

Assume that $\gamma$ is a starlike Jordan curve in the complex plane w.r.t. 0. Let $\alpha\in (0,\pi/2]$. For each $z\in\gamma$, $z\neq x$, we let $\alpha(z,x)$ denote the acute angle which the ...
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81 views

Is there a metric defined on the product space of orthogonal groups?

If one considers just the orthogonal group, then there is a natural metric given by [1]: $$\begin{align} \theta & =\frac{1}{2} \| \log(R_1^{-1}R_2) \| \\ & = \frac{1}{2} \| ...
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1answer
207 views

Generalization of Stewart's theorem?

I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I ...
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56 views

What is the minimal number of lines needed to partition a simplex into cells of diameter at most $\epsilon$?

I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines ...
6
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1answer
95 views

metric condition forcing convex position

Let $A_1,A_2,\ldots, A_n$ be distinct in the plane. For every $1\le i \le n$, let $S_i=\sum\limits_{j=1}^n d(A_i,A_j)$ be the sum of distances from $A_i$ to all the other points. Assume that ...
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1answer
130 views

a property implying co-circularity

Let $A_1, A_2,\ldots, A_n$ be $n$ distinct points in the plane. For every $1\le i\le n$, let $D_i$ be the sum of the distances from point $A_i$ to all the other points. Suppose that $D_i=D_j$ for ...
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120 views

Boomerangs in Polya's orchard

Polya's orchard problem asks for what radius $r$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard. The answer is known; $r$ ...
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1answer
112 views

When a Riemannian manifold with boundary is an Alexandrov space?

Let $M$ be a smooth Riemannian manifold (without boundary). Let $X\subset M$ be a smooth compact submanifold with boundary, $\dim X=\dim M$. Under what conditions $X$, equipped with the induced ...
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1answer
62 views

Existence or otherwise of a set of “sufficiently intricate” open cells

In my question Existence or otherwise of a set of "sufficiently intricate" open sets, I asked about whether it is possible to partition Lebesgue-almost all of $\mathbb{R}^d$ into a finite ...
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1answer
113 views

Closest point to a dual lattice point (in particular for root lattices!)

Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with ...
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1answer
100 views

Existence or otherwise of a set of “sufficiently intricate” open sets

Fix $d \in \mathbb{N}$. Do there exist mutually disjoint connected open sets $V_1,\ldots,V_n \subset \mathbb{R}^d$ and $\mathbf{v} \in \mathbb{R}^d$ such that $\mathbb{R}^d \setminus ...
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72 views

What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?

Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...
5
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113 views

Volume growth of balls

Let $G$ be a locally compact group and $K\subset G$ a compact subgroup. Suppose that on the homogeneous space $X=G/K$ we have a $G$-invariant proper metric $d$. For $R>0$ let $B(R)$ be the open ...
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1answer
92 views

Lamination as limit of arcs

I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...
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1answer
147 views

Classification of 2-dimensional Alexandrov spaces

Is it possible to classify explicitly compact 2-dimensional Alexandrov spaces with curvature bounded below (either with or without boundary)? If yes, a reference would be helpful. EDIT: If the ...
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2answers
263 views

Which surfaces admit unbounded-length simple geodesics?

Let $S$ be a surface embedded in $\mathbb{R}^3$. A simple geodesic on $S$ is one that does not self-intersect. Some surfaces have simple geodesics whose length exceeds any given bound $L$. For ...
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158 views

Distance between quadratic forms

In notes here http://math.univ-lyon1.fr/homes-www/gille/prenotes/lens.pdf on page $2$ a formulation of distance between two positive quadratic form $[q],[q']$ is given by ...
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1answer
109 views

Geometry of convex subsets in Alexandrov space/ Riemannian manifold

Let $X^n$ be an $n$-dimensional complete Alexandrov space with curvature bounded below (or a smooth Riemannian manifold, possibly with boundary). Let $U\subset X$ be an open dense subset with the ...
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1answer
134 views

Are the primary parallelotopes classified? (equivalently, Voronoi cells of lattices)

A primary parallelohedron is a polyhedron that can fill space with infinite translated copies. It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, ...
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1answer
68 views

Optimal covering with finite subcollection of open sets

This is mainly a reference request. Consider a finite collection of (let's say, for simplicity) of open balls $B_i, i = 1, 2, ..., m$ in (again, for simplicity) $\mathbb{R}^n$. I am looking for ...
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2answers
178 views

Volume of the convex hull of the set of all graphic sequences of a given length

Consider the set of all graphic sequences with $n$ elements as a subset of $\mathbb{R}^{n}$, namely let $$D(n)=\{(d_{1},\dots,d_{n})\in\mathbb{Z}_{+}^{n}:d_{1}\geq\dots\geq d_{n},\ ...
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1answer
75 views

Convergence of functions on Alexandrov spaces

Consider a sequence of $n-$dim Alexandrov spaces with curvature $\geq$ -1 $\{(M_i,p_i)\}$ Gromov-Hausdroff converging to an $n-$dim Alexandrov space $(M,p)$. Let $f:M\mapsto \mathbb R$ be a Lipschitz ...
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1answer
67 views

A property of concave functions on Alexandrov spaces

EDIT: Let $X$ be an $n$-dimensional Alexandrov space with curvature bounded below. Let $f_1,\dots, f_n\colon X\to \mathbb{R}$ be $\lambda$-concave functions. Assume that at a fixed point $p$ there ...
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1answer
58 views

A bound on the Haussdorff distance

Let $X, Y \subset \mathbb{Z}^2$ be two discrete and bounded sets. Let $f_X$ be the Euclidean signed distance function of $X$ (similarly for $Y$) and $d_H(X,Y)$ the Euclidean Haussdorff distance ...
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1answer
98 views

Reference: Finsler Derivative?

On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...
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1answer
98 views

Countable subcover of half-open cylinders

While preparing a lecture on dynamic programming principle in optimal stochastic control after the book of Touzi, I discovered a gap in the proof of DPP (page 28 of the book). Here I simplify the ...
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2answers
200 views

Geodesic on Banach Manifold [closed]

Is there a way of defining a geodesic on a Banach Manifold $M$ which is not itself a Hilbert Manifold?
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1answer
100 views

Why simple closed curves are dense in $\mathcal{PML}_0(S)$?

I have another question about laminations on surfaces. As usual let $\mathcal{S}$ be the set of homotopy classes of simple closed curves in $S$ and $\mathcal{PML}_0(S)$ be the set of projective ...
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0answers
67 views

Shortest paths stepping on rational points of height $h$

Q. Do shortest paths walking between rational points of height $h$ ever properly cross themselves? Explaining this question takes a bit of definitional exposition. First, I copy definitions ...
3
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1answer
121 views

Why is $\mathcal{PML}_0(S)$ compact?

I'm starting to study geodesic laminations on hyperbolic surfaces and in particular I'm focusing my attention on $\mathcal{PML}_0(S)$, the space of projective classes of measured geodesic laminations ...
2
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1answer
80 views

Coordinate chart of concave functions near a regular point in Alexandrov spaces

Let $M$ be an Alexandrov space with curvature $\geqslant -1$. Then we have the following theorem which is often used to perturb a regular point to points we want. Let $g_0$ be a ...
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1answer
403 views

Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?

$\newcommand{\til}{\tilde}$ Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds. Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...
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2answers
580 views

Algebraic surface of constant width?

Does there exist an irreducible polynomial $f \in \mathbb{R}[x, y, z]$ such that: $$ V := \{ (x, y, z) \in \mathbb{R}^3 : f(x, y, z) \leq 0 \} $$ is a solid of constant width with a finite symmetry ...
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0answers
60 views

Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...
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1answer
87 views

Gromov-Hausdroff convergence for Alexandrov spaces

Let $\{X_n\}_{n=1}^\infty$ be a sequence of compact Alexandrov spaces (with curvature $\geq k$) converging to (in the sense of Gromov-Hausdroff convergence) an Alexandrov spaces $X$, and ...
3
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1answer
147 views

How many points are in such set with the same norm-2

Let $L=[a,b]\cap\mathbb{N}$ with $a,b\in\mathbb{N}$, let $D\in\mathbb{N}$, and let $C=L^D$. Then I would like to know how many points are there in $C$ with the same given norm-2 $d$. I.e., I'm looking ...
3
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1answer
124 views

How to show it is contained in a convex hull?

There are $(d+1)f$ points (denote the set of all points as $S$) in $\mathbb{R}^d$, that can be divide into $d+1$ disjoint sets $F_1,...,F_{d+1}$, each set of size $f$. If we have $$ ...
8
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2answers
297 views

Constructing a function over a metric space through given points

Suppose there is a compact metric space $(X,\rho)$ and a Euclidean space $\mathbb{R}^n$. There is a sequence of unequal points $\{x_1,...x_N\}$ in $X$ such that all metrics $\rho(x_i,x_j)$ are known ...
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0answers
118 views

GPS calculations under $L^p$ norms

GPS calculations require finding a sphere externally tangent to four given spheres, an Apollonian problem in $\mathbb{R}^3$. The center of that fifth sphere is one of the $16$ possible solutions to ...
4
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1answer
201 views

Combinatorial description of a 120-cell

I'd like a combinatorial description of the 1-skeleton of the 120-cell (roughly) along the lines of the following description of the 1-skeleton of a dodecahedron. (View all elements of product sets ...
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109 views

Connectedness of cones in the boundary of a 1-ended hyperbolic group

Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...
6
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1answer
116 views

Self-avoiding/reflecting geodesics on a convex surface

Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$. For me $S$ is a convex polyhedron, but I am happy to view $S$ as a smooth body with positive Gaussian curvature at each point, or ...
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1answer
330 views

Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle? You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...
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1answer
121 views

what's the formula of the inradius of a general simplex? [closed]

As the title, I just want to know whether there is a general formula for calculating the inradius of a n-simplex. Thank you!
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5answers
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Tiling the plane with incongruent isosceles triangles

It is not difficult to tile the plane with incongruent triangles. One could tile with equilateral triangles, and then partition each equilateral into three triangles, displacing their common ...
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1answer
181 views

Convergence in the proof of Crofton's Formula

Let $\mathcal{L}$ be the set of oriented lines in $\mathbb{R}^2$ and let $\mu$ be the Kinematic Measure on $\mathcal{L}$; up to scaling, $\mu$ comes from the unique (up to scaling) volume form on ...
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454 views

Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$? There are some results of the ...
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1answer
156 views

Intersection of rotating regular polygons

This question has a recreational flavor, but may not be entirely uninteresting. Let $P_k$ be a unit-radius regular polygon of $k$ sides, and $P_n$ a unit-radius regular polygon of $n \ge k$ sides. ...
5
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1answer
76 views

Determining the stretch of a cluster of points

I am trying to determine a metric for measuring cluster stretch. Let C be a cluster of points P0,P1,...,Pn in a two dimensional space with same units. I need a metric that will allow me to ...
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2answers
287 views

Motivation for Hirzebruch-Jung Modified Euclidean Algorithm

Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows: Let $e_i \in \mathbb{N} >2$, and $ r_k \in ...