Tagged Questions

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

learn more… | top users | synonyms (1)

7
votes
1answer
533 views

How large can you draw an island on a map?

A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...
22
votes
1answer
1k views

The view from inside of a mirrored tetrahedron

Suppose you were standing inside a regular tetrahedron $T$ whose internal face surfaces were perfect mirrors. Let's assume $T$'s height is $3{\times}$ yours, so that your eye is roughly at the ...
13
votes
1answer
260 views

Is there a bounded sequence of points in the plane with pairwise distances at least $1/\sqrt{|i-j|}$?

Previously I have mentioned the following problem in an addition to the list of Contest problems with connections to deeper mathematics.. Is there an infinite bounded sequence $(P_n) \subset ...
-4
votes
0answers
60 views

Polyhedron and Euclidean space [closed]

If a point P in the interior of a convex polyhedron in Euclidean space is called a pivot point of the polyhedron if every line through P contains exactly 0 or 2 vertices of the polyhedron. What is the ...
0
votes
0answers
19 views

Literature on ellipses [migrated]

I'm looking for good literature/sources that I can cite for working with ellipses, particularly concerning the overlap of 2 or more ellipses but also other calculations with ellipses (e.g. dividing ...
26
votes
2answers
583 views

Term for “uncheckable constructions”

Is there a term for "uncheckable geometric constructions"? Say, Angle Trisection and Doubling the Cube are checkable; i.e., if the answer is given one can do finite Compass-and-straightedge ...
1
vote
1answer
208 views

Word metrics and finite index subgroups

Suppose that we are given some finitely generated group $ G $ and some finite index subgroup of it $ H $. Given a finite generating symmetric generating set $ S \subset G $, we can define the word ...
4
votes
1answer
187 views

Panning for gold nuggets: a type of isoperimetric problem

Let $C$ be a unit-radius circle in the plane. Suppose you have a total length $L$ of string available, and your task is to connect chords of $C$ using no more than $L$ of string to minimize the ...
14
votes
1answer
206 views

Does the boundary of a convex body contain a regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ includes 5 points which form a regular planar pentagon? The following consideration suggests the answer "yes": if we ...
3
votes
1answer
74 views

Stable equilibria of points on the 2-sphere

Suppose $n$ points lie on the sphere $S^2=\{x\in\mathbb{R}^3\mid \|x\|=1\}$ and are subjected to a repulsive acceleration that pushes away a point from each other point with an intensity proportional ...
1
vote
0answers
34 views

Projection from a polytope to an affine space

Let $P\subseteq \mathbf{R}^n$ be some polytope defined by an intersection of half spaces with corresponding hyperplanes $H_k$, and let $A\subseteq \mathbf{R}^n$ be some affine space, with $A\cap P ...
4
votes
1answer
124 views

The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher

According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...
5
votes
0answers
66 views

Path metrics without geodesics [duplicate]

This is a follow-up of this question. Recall that a metric space $(X,d)$ is called a path-metric space if the distance between any two points in $X$ equals the infimum of lengths of paths between ...
3
votes
1answer
109 views

Three-dimensional Apollonian spirals

Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$. Let ...
6
votes
1answer
310 views

Reverse plane geometry, anyone?

I refer to Greenberg's wonderful 2010 MAA article "Old and new results in the foundations of elementary plane Euclidean and non-Euclidean geometries". There, and in his book, Greenberg defines a ...
-1
votes
1answer
197 views

Creating topological spaces with portals [closed]

I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension. I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
6
votes
4answers
171 views

Inside-out polygonal dissections

A dissection of a polygon $P$ is a partition of $P$ into a finite number of pieces, which can then be rearranged (via planar translations and rotations) and joined (without overlap) to form a new ...
6
votes
2answers
224 views

Counting valid coordinates

We are given a matrix $D = (d(i,j))_{1 \leq i,j \leq n}$ such that $d(x,z) \leq d(x,y) + d(y,z)$ for each $1 \leq x,y,z \leq n$. It is also known that $d(x,y) \in \mathbb{N}$ (In this question $0 \in ...
0
votes
2answers
151 views

Determine the boundary points of a set of points [closed]

I have a set of points $S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$. Then how to find the boundary points (which is a subset of $S$) of $S$? There are methods like convex hull, concave hull and ...
3
votes
0answers
56 views

Probability that a random projection doesn't reduce the distance of a point from a subspace too much

Consider the natural uniform measure (is it called the Haar measure?) on the set of $(n-k)$-dimensional subspaces of $R^n$. We are given a $d$-dimensional affine subspace $U$ (think of $d, k \ll n$; ...
14
votes
2answers
1k views

“a shape that … lies halfway between a square and a circle”

An article in the Notices of the AMS, Volume 61, Issue 10, 2014 (PDF download link), on Khot's Unique Games Conjecture, says this: Another group ... found a shape that in a certain sense lies ...
0
votes
0answers
40 views

How to show the Poincaré disk is hyperbolic for some $\delta$ [migrated]

I am trying to prove that the Poincaré disk, $\mathbb{D}$, is $\delta$-hyperbolic with respect to the slim triangle definition for hyperbolicity. I have been stuck for a while on where to begin, ...
4
votes
0answers
106 views

Optimal planar net for catching convex shapes

Imagine you want to make a net out of string to filter and catch objects of a certain size, minimizing the length of string employed. (This actually arises in filtering biological impurities from ...
6
votes
2answers
311 views

Are angles between points enough to decide the realizability?

Let n points in the plane be given whose coordinates we don't know. Assume, however, that for any triple of the points we know the angle. Question: Can we decide whether the n points are realizable ...
6
votes
0answers
115 views

A forked plane continuum

I came up with this question while trying to solve the following MO one: Does every connected set that is not a line segment cross some dyadic square? Suppose $C$ is a plane continuum (i.e. a ...
0
votes
0answers
166 views

A question about Assaf Naor's review in Bourbaki about the Batson-Spielman-Srivastava result

I am referring to this article - http://www.cims.nyu.edu/~naor/homepage%20files/Exp.1033.pdf If I understand right, the author states that his equations (8) and (9) are equivalent to the equations ...
1
vote
2answers
58 views

Generalised “projection” of a metric space

Assume we have $n$ points $p_0\ldots p_{n-1}$ which form a discrete metric space $V$ with metric $d$. Can we define a function $f:V\rightarrow \mathbb{R}$ with $f(p_0) = 0$, $f(p_1) = d(p_0,p_1)$ and ...
2
votes
1answer
64 views

Visibility kernels of embedded graphs

Let $G$ be a connected graph embedded in the plane with all edges straight segments. For $\alpha \in (0,\pi)$, define an $\alpha$-path as a path in $G$ with all turns at vertices within ...
2
votes
0answers
105 views

Better Sobolev inequality holds in this case when assuming doubling and Poincare inequality?

Let $X$ be a Polish space and let $m$ be a locally finite Borel measure on $X$. Let $\epsilon$ be a strongly local, regular Dirichlet form on $L^2(X,m)$ with Domain $V :=\{f\in ...
0
votes
0answers
51 views

Intersection points of closed curves inscribed in a convex polygon

Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...
2
votes
1answer
69 views

How to infer missing nodes from a path?

I have a first data set which is a list of train stops with coordinates (lat, lon), but not the "links" between the nodes/stops (this could thought of as a null or empty graph). I have a second data ...
14
votes
4answers
402 views

The limit of edge-midpoint convex polyhedra

    Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$, replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$. Continuing this process, we obtain a ...
1
vote
0answers
440 views

A metric on $S^{2}$ [closed]

Edit:Can this new version of this question be answered with the method of same comments to the previous version? Let $p:S^{3}\to S^{2}$ be the Hopf fibration $p(z,w)= (\parallel ...
8
votes
3answers
508 views

Separating points in the plane II

Let A be a set of $2m$ points on the plane so that no open set of diameter $2$ has more than m of them. Define $A+A+...+A$ ($k$ times) to be the multiset of $k$-sums from $A$. That is, we consider all ...
7
votes
2answers
313 views

Rep-tiles of order 2

A 2-rep-tile is a geometric shape that can be partitioned into exactly 2 smaller (dilated) copies of itself. Although there are many rep-tiles of higher orders, the only 2-rep-tiles I could find ...
2
votes
1answer
65 views

Triangulations of translation surfaces whose edges are shorter than the diameter

Let $S$ be a compact translation surface (i.e. a surface endowed with a singular flat metric such that singular points are locally isometric to a cone of angle an integer multiple of $2\pi$, and that ...
-1
votes
1answer
82 views

CAT spaces and Metric Measure Spaces [closed]

This is very vague question but here it goes: are there any results of analysis on metric spaces in CAT spaces?, for instance as in the paper of Sturm (On the geometry of metric measure spaces) or as ...
39
votes
2answers
2k views

Randall Munroe's Lost Immortals

In Randall Munroe's book What If?, the "Lost Immortals" question asks: If two immortal people were placed on opposite sides of an uninhabited Earthlike planet, how long would it take them to find ...
3
votes
0answers
103 views

Surfaces ruled through a subset of points

One common definition of a ruled surface $S$ is that, through every point $p \in S$, there passes a line $L(p)$ that lies in $S$: $L(p) \subset S$. My question is: Q0. Is there any loosening of ...
6
votes
1answer
205 views

Thinnest 2-fold coverings of the plane by congruent convex shapes

It is an unsolved problem to determine the "thinnest" $2$-fold covering of the plane by disks. The $2$-fold coverage problem by disks is to find the minimum number of congruent (unit-radius) disks ...
8
votes
1answer
153 views

Gromov-Hausdorff convergence for non-compact metric spaces

Let $(X_i,p_i)$, $(X,p)$ be pointed connected proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent? $\forall r > 0: \bar{B}_r(p_i) ...
3
votes
1answer
136 views

Random non-intersecting circles in the plane

If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...
2
votes
0answers
39 views

Surfaces with curvature $\leq K$ are of bounded integral curvature

One characteristic of a CBA($K$) surface (a topological surface with an intrinsic metric of curvature $\leq K$ in the sense of Alexandrov) is that $\delta_K(T) \leq 0$, where $\delta_K(T)$ is the ...
0
votes
1answer
101 views

Generalization of the triple tangent identity

It is well known that if $x + y + z = \pi$ then $$\tan x \times \tan y \times \tan z = \tan x+ \tan +\tan z.$$ I came across the following generalization of this equality: $$\sqrt{1-k^2} {\rm ...
3
votes
0answers
54 views

A taut string of equilateral triangles

Let $T$ be a unit edge-length equilateral triangle composed of three cylinders each of (small) radius $r>0$. (By "small" I mean approximately $< 0.1$.) Think of $T$ as a physical, rigid ...
3
votes
1answer
88 views

Does this squared distance functional have a unique critical point on geodesically convex manifolds?

Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow ...
3
votes
0answers
68 views

Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...
7
votes
3answers
320 views

The distribution of the shortest path through $n$ points

In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small? More ...
5
votes
0answers
169 views

Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?

Recent questions showed that roots of a random polynomial tend to lie on the unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
5
votes
1answer
203 views

Maximum number of general-position points with mutual rational distances?

Richard Guy has shown that there are six points in the plane—no three collinear, no four cocircular—such that all interpoint distances are rational. Guy, Richard. Unsolved Problems in ...