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

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5
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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
127 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
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1answer
331 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
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1answer
211 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
195 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
234 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 ...
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2answers
241 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
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0answers
66 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 $d, k \ll n$; ...
16
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 ...
4
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0answers
113 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
324 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
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0answers
122 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
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0answers
182 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
67 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
66 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
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0answers
117 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
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0answers
66 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
73 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
434 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 ...
8
votes
3answers
521 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
340 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
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1answer
70 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 ...
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votes
1answer
95 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
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0answers
104 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
232 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
198 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
178 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? ...
3
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0answers
78 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 ...
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1answer
173 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
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0answers
72 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
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1answer
106 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
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0answers
81 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
337 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
178 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
215 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 ...
6
votes
1answer
122 views

Translative packing constant strictly larger than lattice packing constant

Simply put, my question is this: what is the smallest dimension, if any, where we can know for sure that a convex body exists whose translative packing constant is strictly larger than its lattice ...
4
votes
3answers
265 views

Peeling a polygonal vegetable

When you peel a vegetable, such as a potato or a cucumber, you usually remove its head, then contiually remove parts of its skin, until you remain with the pulp alone. I would like to formalize this ...
33
votes
3answers
924 views

What polygons can be shrunk into themselves?

Let's call a polygon $P$ shrinkable if any down-scaled (dilated) version of $P$ can be translated into $P$. For example, the following triangle is shrinkable (the original polygon is green, the ...
8
votes
1answer
225 views

Dubins car shortest paths: Decidable?

A Dubins car follows a Dubins path in $\mathbb{R}^2$, with constant wheel speed and limited turning radius. It is known that the shortest Dubins path in the absence of obstacles follows circular arcs ...
3
votes
1answer
220 views

Dirichlet polyhedra for hyperbolic manifolds

Let $H$ be a simply-connected, complete space of constant negative curvature, that is, a hyperbolic space, $\Gamma$ a discrete group of isometries, and and $M=H/\Gamma$ its quotient space; we assume ...
6
votes
0answers
111 views

CAT(0) groups that does not act on CAT(0) cubical complex

CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...
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votes
2answers
593 views

what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics

In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of ...
6
votes
4answers
359 views

What can be said of the structure of a metric space without isosceles triangles?

This is a question that has been bothering me in the back of my head for quite some time. Suppose we have a metric space $X$ with metric $\mathrm{d}$. By an isosceles triangle we mean a tuple of ...
7
votes
2answers
122 views

Trees with a maximal convex hull: are the only optimal solutions Steiner trees?

For given $n\geqslant 3$, I'm looking for a connected set composed of $n$ equal segments in the plane such that the convex hull of it has maximal area $A(n)$. To simplify notation, we'll take ...
4
votes
0answers
78 views

Nice minimal embeddings of large finite groups into compact Riemannian manifolds

The initial motivation for this question is a very practical problem: I need to find the absolute minimum of a function on a very large symmetric group $\Sigma_N$ (with $N$ 10000 or more). So if ...
15
votes
1answer
247 views

Higher dimensional generalization of: Any quadrilateral tiles the plane?

Any (non-self-intersecting) quadrilateral tiles the plane.     (MathWorld image.) Q. What is the strongest known generalization of this statement to higher dimensions? I.e., ...
18
votes
2answers
523 views

Is every elementary absolute geometry Euclidean or hyperbolic?

Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary ...
11
votes
1answer
300 views

Simple, closed geodesics in $\mathbb{S}^3$ manifold

Lyusternik and Shnirel'man were the first to prove Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has at least three simple (non-self-intersecting), closed geodesics. See, e.g., ...
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0answers
118 views

Circumscribing simplex to convex body?

Q. Does every (perhaps smooth) compact convex body $K$ in $\mathbb{R}^d$ admit a circumscribing simplex, each facet of which touches (shares a point with) $K$? How about a circumscribing ...