**5**

votes

**1**answer

165 views

### Non-closed geodesics on a convex polyhedron in $\mathbb{R}^3$

Let $P$ be the surface of a closed convex polyhedron in $\mathbb{R}^3$.
Q. Does every non-closed geodesic $\gamma$ fill $P$ densely?
Of course $\gamma$ cannot pass through a vertex of $P$, but ...

**6**

votes

**3**answers

197 views

### Most dispersed set of points in a disk?

Put $1$ billion points in a disk of radius $1$. Consider the minimal area $A$ of a triangle formed by any $3$ points. Where do you put the points so that $A$ is maximal and how much is $A$?
Consider ...

**11**

votes

**3**answers

343 views

### Orthogonal mud cracks and Maxwell's reciprocal figures

Is there a succinct mathematical/physical explanation of why mud cracks
tend to meet orthogonally?
Wikipedia image in this ...

**1**

vote

**0**answers

192 views

### Find m most distant points from a set of n points [closed]

I would like to find the $m$ (where $n$ $\geq$ $m$ > 1) maximally distant subset of points from a collection of $n$ $d$-dimensional points. Maximally distant means the sum of the pairwise distances ...

**1**

vote

**1**answer

174 views

### Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle

Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much.
Let $M$ be a n dimensional manifold. ...

**5**

votes

**2**answers

359 views

### Generalization of Pascal's Theorem to Higher Dimensions

Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement ...

**3**

votes

**1**answer

75 views

### Fixed points of finite order isometries of metric spaces

I would like to show the following:
Let $X$ be a complete metric space that is uniquely geodesic (i.e. each two distinct points are connected by a unique geodesic segment) and $\phi\colon X\to X$ an ...

**3**

votes

**0**answers

46 views

### Fixed point of fatness

For the purposes of this question, define the following properties of convex sets in the plane:
A set is $R$-fat (for $R\geq 1$) if it contains a disc of side-length $x$ and is contained in a disc ...

**2**

votes

**0**answers

83 views

### Besicovitch's covering theorem for ellipsoids and shadows

The usual Besicovitch's covering theorem concerns closed balls in $\mathbb{R}^d$. It relies on a property called "directionally limited metric space": the principal ingredient is to say that there ...

**9**

votes

**0**answers

282 views

### How can we join two points with a small ruler? [closed]

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this? Imagine a method that works when $AB$ is really huge and ...

**4**

votes

**1**answer

153 views

### Maps between spaces of non-empty compact subsets with the Hausdorff distance (reference request)

Let $X, Y$ be metric spaces, and let $PX$ (resp. $PY$) be the set of all non-empty compact subsets of $X$ (resp. $Y$) with the Hausdorff metric. A continuous map $f\colon X\to Y$ induces a continuous ...

**5**

votes

**2**answers

317 views

### Equitably distributed curve on a sphere

Let $\gamma=\gamma(L)$ be a
simple (non-self-intersecting) closed curve of length $L$
on the unit-radius sphere $S$.
So if $L=2\pi$, $\gamma$ could be a great circle.
I am seeking the most equitably ...

**17**

votes

**0**answers

287 views

### Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set ...

**8**

votes

**1**answer

708 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 ...

**43**

votes

**2**answers

2k 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 ...

**14**

votes

**1**answer

345 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 ...

**28**

votes

**2**answers

678 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

**1**answer

570 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

**1**answer

206 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 ...

**16**

votes

**1**answer

323 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

**1**answer

85 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

**0**answers

49 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

**1**answer

185 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

**0**answers

71 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

**1**answer

129 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

**1**answer

339 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

**1**answer

215 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

**4**answers

211 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

**2**answers

237 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

**2**answers

376 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

**0**answers

70 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

**2**answers

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**

votes

**0**answers

116 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

**2**answers

335 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 ...

**7**

votes

**0**answers

134 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

**0**answers

191 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

**2**answers

70 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

**1**answer

67 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

**0**answers

121 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

**0**answers

77 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

**1**answer

76 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

**4**answers

450 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

**3**answers

523 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

**2**answers

354 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

**1**answer

75 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

**1**answer

101 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

**2**answers

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

**0**answers

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

**1**answer

236 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

**1**answer

224 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) ...