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

learn more… | top users | synonyms

1
vote
0answers
105 views

When is the median closest nearest-neighbor distance larger than the mean closest nearest-neighbor distance?

Consider a random Poisson process in an $d$-dimensional cube of arbitrary size (alternatively, consider an arbitrarily large $(d-1)$-dimensional sphere in an $d$-dimensional space). If we have a ...
10
votes
3answers
637 views

A special tessellation

Let $P$ be a convex $n$-gon. Suppose that we have an infinite number of $P$s, and that each of them is colored either red or blue. Here, let us consider the following operations : Operation 1 : Place ...
10
votes
1answer
227 views

Mapping class group and CAT(0) spaces

I hope the questions are not too vague. Is the mapping class group of an orientable punctured surface $CAT(0)$ ? Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...
4
votes
0answers
99 views

Fourier analysis for the discrete cube in CAT(0) spaces?

Is there a meaningful Fourier analysis of mappings from the discrete cube into CAT(0) spaces? Examples for what I have in mind: Fix a CAT(0) space $X$, a mapping $f:\{-1,1\}^n \to X$, and ...
2
votes
1answer
101 views

Space of simple polygons on $n$-vertices as a set of points in $\mathbb{R}^{2n}$

A simple polygons in $\mathbb{R}^2$ with $n$ vertices can be mapped to elements in $\mathbb{R}^{2n}$ by list the coordinates of it's vertices. I expect there might be something interesting to study ...
-2
votes
1answer
213 views

Polygon Problem [closed]

There are $N$ regions which are numbered from $1$ to $N$. Each region is represented by a single simple polygon on the 2D plane. Simple polygon means the boundary of the polygon does not cross itself. ...
6
votes
0answers
119 views

Variations on a problem of S. Mazur

In problem 76 of the Scottish Book Mazur asked Given a convex body $K$ in three-dimensional space and a point $o$ in its interior, consider the surface $S$ formed by all points $p$ such that the ...
2
votes
2answers
131 views

Largest inscribed rectangle inside a convex polygon

It has been proved by Radziszewski in this paper K. Radziszewski. Sur une probleme extremal relatif aux gures inscrites et circonscrites aux gures convexes. Ann. Univ. Mariae Curie-Sklodowska, Sect. ...
1
vote
1answer
96 views

Does John's Ellipsoid preserve subset ordering? [duplicate]

Let $K \subset \mathbb{R}^d$ be a convex body, symmetric about the origin and with nonempty interior. Then John's theorem asserts that there exists a unique ellipsoid $E$ of minimal volume such that ...
11
votes
3answers
241 views

Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$?

The geometrization theorem tells us: Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a ...
20
votes
7answers
693 views

Is there always a maximum anti-rectangle with a corner square?

Let $C$ be an axis-parallel orthogonal polygon with a finite number of sides. Define an anti-rectangle in $C$ as a set of small squares in $C$, such that no two of them are covered by a single large ...
0
votes
0answers
58 views

Minimum distance larger than a fraction $f$ of the closest nearest-neighbor distances for points placed by a random Poisson process?

Consider a random Poisson process on arbitrarily large volume in $R^d$ enclosed by an $R^{(d-1)}$ dimensional sphere. The process terminates when a density of points $\rho$ is achieved (letting $N$ ...
5
votes
3answers
226 views

When does heat kernel have both Gaussian upper and lower bounds?

Recently, I am reading Sturm's paper "Analysis on local Dirichlet form III: X is a locally compact separable Hausdorff space and m is a positive Radon measure with supp[m]=X. $\varepsilon$ is a ...
1
vote
0answers
77 views

method to construct platonic solids from the circumscribed sphere [closed]

I am trying to find a geometrical method of building the platonic solids starting from the circumscribed sphere. I do have a class of generative 3d modeling where we create 3d object programmatically ...
12
votes
3answers
327 views

Embedding expanders in CAT(0) spaces

It is well-known that expanders are hard to embed into Hilbert (or $\ell^p$) spaces - any embedding of an expander with $n$ vertices has distortion $\Omega(\log n)$. Can anyone provide a reference ...
19
votes
1answer
248 views

Hidden points in polygons

Let $h(n)$ be the largest number of mutually invisible points that can be located in a polygon $P$ of $n$ vertices. Two points $x$ and $y$ are mutually invisible if the segment $xy$ contains a point ...
14
votes
1answer
518 views

Infinite desert with waterpoints

UPDATE: I created a simple web-application, that allows the user to move waterpoints around, then automatically calculates a maximum set of interior-disjoint squares between the points (the algorithm ...
1
vote
1answer
124 views

heat kernel $p_t(x_0,y) \in D(\Delta) \cap L^\infty$ for a manifold with Ricci curvature bounded below?

X is an n-dim Riemannian manifold with the Dirichlet form $$ \varepsilon (u,v) =-\int_X \langle \nabla u,\nabla v \rangle $$ for $u,v \in W^{1,2}(X)$. Let $P_t$ and $p_t(x,y)$ be the associate ...
2
votes
0answers
77 views

n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
3
votes
2answers
118 views

Lattice-point-free buffers around circles

Let $C(r)$ be the origin-centered circle of radius $r$, and let $\beta(r)$ be the exterior buffer around $C(r)$: the distance from $C(r)$ to the closest lattice point exterior to $C(r)$: ...
2
votes
3answers
295 views

Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?

For any convex $n$-gon $P_{0,1}P_{0,2}\cdots P_{0,n}$, let us consider the following operation : Operation : Let $k=0,1,\cdots$. Take $n$ points $P_{k+1,i}\ (i=1,2,\cdots,n)$ outside of $n$-gon ...
3
votes
1answer
117 views

About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true? "Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of ...
7
votes
0answers
141 views

Lattice radial-step (ratchet) spirals

(30Oct13: Now solved; see Addendum.) Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.     $S(r_0,\epsilon)$ begins with the arc ...
0
votes
0answers
62 views

$|\nabla f|^2 \in W^{1,2}(M)$ for harmonic function f on manifolds with two nonparabolic ends?

We know that if a complete noncompact manifold M has two nonparabolic ends, then we can construct a nonconstant bounded harmonic function with finite Dirichlet integral defined on the whole M. ...
1
vote
0answers
63 views

Is the size of $\varepsilon$-nets of the Euclidean ball exponential for large $\varepsilon$

Let $X$ be the unit ball of $(\mathbb{R}^n,\|\cdot\|_2)$. A finite set $N=N(\varepsilon)$ is a $\varepsilon$-net of $X$ if every point in $X$ is at most a distance $\varepsilon$ from a point from $N$. ...
2
votes
1answer
108 views

Are harmonic maps quasiconformal at the boundary of hyperbolic spaces?

Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$. If $\rho:\pi_1M\rightarrow Isom(X)$ does not fix a point in $\partial ...
4
votes
1answer
170 views

A construction related to scissors congruence

I was thinking about the following some time ago. My question is whether such things have been studied before. Let $E_n$ be the abelian group with a generator for each (bounded) euclidean polytope of ...
3
votes
0answers
73 views

The Tangent Bundle of the Space of CR Structures on S^(2n+1)

Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...
2
votes
2answers
222 views

what's the best way to characterise the distribution of prime elements in simple perfect squared squares

DEFINITIONS: A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the ...
4
votes
1answer
77 views

generator of Dirichlet form coincide with the absolute part of the “Laplacian”

Let M be an Riemannian manifold with the Dirichlet form $$\varepsilon (u,v) =-\int_M \langle \nabla u,\nabla v \rangle$$ for $u,v \in W^{1,2}_0(M)$. Let $\Delta^M:D(\Delta^M) \to L^2(M)$ denote the ...
15
votes
2answers
269 views

Are there locally jammed arrangements of spheres of zero density?

I know of a remarkable result from a paper of Matthew Kahle (PDF download), that there are arbitrarily low-density jammed packings of congruent disks in $\mathbb{R}^2$: In 1964 Böröczky used a ...
4
votes
1answer
117 views

Maximal geometric mean of distances between points on an interval

Suppose I had T points in the interval $[0,1]$. Call them $e_1, \dots, e_T$. Question 1: What is a good nontrivial bound on the geometric mean of $$\{|e_i - e_j| : 1 \leq i < j \leq T \}, $$ as a ...
1
vote
1answer
61 views

Dirichlet energy with domain $W^{1,2}(M)$ or $W^{1,2}_{loc}(M)$ can be a specific Dirichlet form?

M is a Riemannian manifold, $\varepsilon(f,g)=\int_M \langle {\nabla f,\nabla g}\rangle dvol$. Then with which domain is $\varepsilon$ a strongly local, regular and tight Dirichlet form? $W^{1,2}(M)$ ...
22
votes
3answers
649 views

Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:       Starting with any triangle $T$, one forms $T'$ by connecting ...
4
votes
1answer
103 views

The space of circular triangles?

A circular triangle is a closed simple curve in the Euclidean plane $\mathbb{R}^2$ that can be expressed as the union of three circular arcs. Data naturally associated with such a figure includes: ...
1
vote
2answers
237 views

How to construct a harmonic function with non-zero gradient on manifold with two nonparabolic ends?

We know that if a complete noncompact manifold M has two nonparabolic ends, then we can construct a nonconstant bounded harmonic function with finite Dirichlet integral defined on the whole $M$. More ...
0
votes
1answer
84 views

Probability of disc-disc overlap for discs placed with uniform probability on a surface until a density $\rho$ is achieved

Imagine I place discs of radius $r$ on a two-dimensional plane, selecting their positions with uniform probability across the surface of the plane, and stop when I reach a disc density $\rho$. As a ...
2
votes
2answers
240 views

Sphere - Symmetry and Triangulation [closed]

The sphere is symmetric with respect to any rotation. However, it loses this property as soon as it is triangulated. Are there sequences of triangulations that possess particular large symmetry groups ...
5
votes
1answer
149 views

Width of a random convex polygon

Consider a planar (2D) random walk comprised of N steps. Consider the minimum convex polygon enclosing the N points visited by the random walker. Assume the definition of the width of a convex ...
1
vote
1answer
55 views

Optimal radiating $(d{-}1)$-flats within a sphere

Permit me to revisit an earlier unresolved MO question, "Chord arrangement that avoids confining small or large disks" with a (very!) specific version, inspired by radiation therapy. The main idea is ...
4
votes
0answers
153 views

Intuition behind minimizing the Dirichlet energy of a mapping

What does minimizing the Dirichlet energy of a mapping $\Phi$ achieve intuitively? Roughly it is the integral (or sum, if discrete) of $|\nabla \Phi(\;)|^2 dV$, with $V$ the volume. So is it, in some ...
5
votes
1answer
155 views

Soddy-type relation for Steiner chains

For Steiner $n$-chains of circles of radii $r_1,\dots,r_n$ tangent to an inner circle of radius $r_-$ and an outer circle of radius $r_+$, is there a Soddy-type relation between the $n+2$ quantities ...
2
votes
2answers
868 views

Heat flow $P_tf \to f$ in $W^{1,2}$ for $f \in W^{1,2}$?

$\varepsilon:L^2(X,m) \to [0,\infty]$ is a strongly local, symmetric Dirichlet form generating a Markov semigroup $(P_t)_{t\ge0}$ in $L^2(X,m)$. Let $D(\varepsilon)=\{f\in ...
-4
votes
1answer
162 views

Space packing fraction of tetrahedron and octahedron [closed]

If four equal size spheres form close-packed tetrahedron, what is the fraction of space that is occupied/unoccupied by the spheres in the tetrahedron. Similarly, if six equal size spheres form ...
7
votes
1answer
439 views

Separating pairs of points in R^n

Let $A$ be a set of $2k$ points in $\mathbb{R}^n$ such that no open set in $\mathbb{R}^n$ of diameter $2$ contains more than $k$ of these points. What is the largest possible distance $r_n>0$ one ...
24
votes
3answers
885 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$. ...
2
votes
0answers
97 views

A few questions about Banach spaces all of whose “points” are infinite sequences of real numbers

I shall call these spaces "sequence spaces". While there exist well known separable Banach spaces that have subsets which are isometric to every separable metric space, I never heard of any separable ...
6
votes
1answer
151 views

Best and worst centrally symmetric convex covering shapes

Suppose you have a centrally symmetric convex 2D shape $C$ of area $A$, and you randomly throw down copies of $C$ on the plane so that each $C$-center lies within a given unit square $S$, until $S$ is ...
1
vote
0answers
72 views

Equidissection of square [duplicate]

Monsky's Theorem states: One cannot dissect a square into an uneven number of triangles with equal area. I was wondering how close one could get to a equidissection, i.e. For $n$ an uneven ...
-1
votes
1answer
106 views

A question about metric spaces that are compact and denumerably infinite

Let S be any compact and denumerably infinite metric space and let d be the metric of S. We shall say that S satisfies condition C, if there exists at least one infinite ...