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

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2
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0answers
111 views

Avoiding Ricci-flow dumbbell neck-pinch by inflating a surface

It is well known that Grayson's dumbbell neck-pinch separates into disconnected pieces under Ricci flow:                     Image source: ...
14
votes
1answer
414 views

Lower-Hölder embeddings of the sphere

My question is very simple: Given $d\ge 3$, does there exist $s\in (0,1)$ and an embedding $f:S^{d-1}\to \mathbb{R}^d$ such that $$ |f(x)-f(y)| \ge |x-y|^s \quad\textrm{if } |x-y|<r, $$ for ...
8
votes
0answers
212 views

Orthonormal bases of R^3 with components lying in the golden field

Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin ...
3
votes
1answer
109 views

Can every large point set be connected to a given knot?

Let $K$ be a given knot, and $P$ a set of points in $\mathbb{R}^3$ in general position, general position in the sense that no three points are collinear and no four coplanar. Define the point-set ...
4
votes
1answer
236 views

Finding the “top” or “bottom” vertex of a simplex

A vertex $v$ of a simplex in $\mathbb{R}^n$ is a "top" vertex if there exists a point $p \neq v$ in the simplex with $v \ge p$ (i.e. $v_1 \ge p_1$, ... , $v_n \ge p_n$). Similarly, $v$ is a "bottom" ...
5
votes
3answers
2k views

Minimum distance between two arbitrary circles in space?

What is the minimum distance between two arbitrary circles in space? I am working out the problem with Maxima, but I am surprised by how complicated this rapidly turns out to unfold for such a ...
8
votes
2answers
847 views

The Gauss circle problem on a hexagonal lattice

Take an infinite hexagonal lattice (or equivalently, an equilateral triangular lattice), with unit spacing between the closest lattice point pairs, and draw a disc of radius $r$ centered on a lattice ...
9
votes
3answers
383 views

Mutually tangent ellipsoids in 3 space

I recently heard a claim that for any n, it is possible to arrange n ellipsoids in 3 space such that each pair of ellipsoids is kissing. Is this true, and if so, how? Edit: By kissing, I mean that I ...
5
votes
2answers
946 views

What is the best *general triangle*?

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...
16
votes
1answer
413 views

Can R^3 be expressed as a disjoint union of pairwise linked circles?

We can express $\mathbb{R}^3$ as a disjoint union of circles. There are some constructive ways of doing this, although it's easier to construct them sequentially by transfinite induction, applying the ...
18
votes
6answers
2k views

Are there smooth bodies of constant width?

The standard Reuleaux triangle is not smooth, but the three points of tangential discontinuity can be smoothed as in the figure below (left), from the Wikipedia article. However, it is unclear (to me) ...
-2
votes
0answers
74 views

Parallel transport along a geodesic and the related Jacobi field [closed]

Crossposted from: http://math.stackexchange.com/questions/1255018/parallel-transport-along-a-geodesic-and-the-related-jacobi-field This is a formula/theorem (written below) that I found mentioned in ...
2
votes
2answers
1k views

Trilateration problem

When trying to develop an algorithm for a program, I got with the following problem: Determine the approximate location of $O$, if you can take finite samples $P_n$ from known locations and always ...
1
vote
1answer
166 views

Construction of an integral point set given the set of distances, its minimal description to get a measure of its complexity and its unique identifier

Given a set of distances between every pair of points of an integral point set $P$ of $n$ points; say $D = \{{d_i}\}$. Q1. What is the least time complexity possible/known for recreating the ...
2
votes
1answer
167 views

Determining A valid Convex Hexagon given The length of Six sides

Suppose We are given the length of all six sides of a Convex Hexagon. How can we tell whether it's valid or Not ? that means can we tell whether it's area is positive or not ?
17
votes
5answers
397 views

How many unit simplices are needed to cover a unit $n$-cube?

The volume of an $n$-dimensional simplex of unit edge length is $$V(n) = \frac{\sqrt{n+1}}{n! 2^{n/2}} \;,$$ so at least $\lceil 1/V(n) \rceil$ such simplices are needed to cover the unit $n$-cube. ...
0
votes
0answers
47 views

On the centroid of a triangle [migrated]

There's three different ways to see a triangle in the Euclidean plane: as three non-collinear points, say $A$, $B$, $C$; as the line segments connecting the three points, that we can parametrize as a ...
7
votes
2answers
172 views

approximate two different real numbers to order $\frac{1}{z^{3/2}}$

I took this result from Minkowski's book on Geometry of numbers: Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and ...
1
vote
1answer
2k views

3D Delaunay Triangulation -> Euclidean Minimum Spanning Tree

I read that the Euclidean Minimum Spanning Tree (EMST) of a set of points is a subgraph of any Delaunay triangulation. Apparently the easiest/fastest way to obtain the EMST is to find the Deluanay ...
2
votes
1answer
77 views

Control of the metric in isothermal coordinates

Suppose you have a riemannian surface $(\Sigma,g)$, and an open simply-connected set $U \subset \Sigma$. You know that you can find isothermal coordinates - that is a map $\varphi : U \rightarrow D$ ...
34
votes
3answers
1k 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 ...
4
votes
1answer
54 views

Largest regular $k$-simplex inscribed in a $d$-cube, $k < d$

The largest (by edge length) regular simplex inscribed in a unit cube is well known in $\mathbb{R}^2$ and $\mathbb{R}^3$:     Image sources: left: NMSU, right: Mathworld. A recent ...
2
votes
2answers
49 views

Volume of a region given by a Constraint Satisfaction Problem

I have a Linear Constraint Satisfaction Problem i.e. I have variables $ x_1, x_2,...,x_m$, with corresponding domains $D_1,D_2,...,D_m $ satisfying linear constraints $C_1, C_2,...,C_n$ with $n ...
11
votes
2answers
256 views

A measure on the space of probability measures

This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
25
votes
6answers
3k views

Covering a unit ball with balls half the radius

This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks": How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of radius ...
17
votes
2answers
450 views

Repeated random two-steps in $\mathbb{R}^3$: unbounded?

I created a random isometry $T$ of $\mathbb{R}^3$ by generating a random orthogonal matrix $M$, uniformly distributed among all such, and a random displacement $v$, whose coordinates are drawn from a ...
1
vote
0answers
26 views

Biggest volume parallelotope inside the union of two parallelotopes

Given a parallelotope $P$ symmetric around the origin, and a vector $v$, such that $(P+v)∩(P−v)$ is not empty, is there a simple way to obtain a parallelotope $Q⊂(P+v)∪(P−v)$, symmetric around the ...
5
votes
2answers
163 views

Biggest parallelogram inside the union of two translated parallelograms

If I have a parallelogram $P$ symmetric around the origin, and a vector $v$, such that $(P+v)\cap (P-v)$ is not empty, is there a simple way to obtain the parallelogram $Q\subset (P+v) \cup (P-v)$, ...
9
votes
1answer
116 views

bi-Lipschitz gluing

Let $H$ be the Heisenberg group with left invariant sub-Riemannian metric and $\varepsilon>0$ is small. Let us denote by $|x-y|_H$ the distance from $x$ to $y$ in $H$. I have a bi-Lipschitz ...
1
vote
1answer
66 views

Alexandrov spaces of constant curvature

Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian ...
26
votes
14answers
5k views

Open problems in Euclidean geometry?

Which are some (research level) open problems in Euclidean geometry ? (Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet) I should clarify a ...
16
votes
4answers
770 views

Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial type of convex polyhedron may be realized by one all of whose edges are tangent to a sphere (and the realization is unique if the center ...
9
votes
3answers
10k views

Dividing a square into 5 equal squares

Can you divide one square paper into five equal squares? You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
4
votes
3answers
310 views

Must the powers of some element always grow linearly with respect to a word metric?

Suppose we have a group $G$ which is finitely generated , and let $|\cdot |$ denote some word metric on it. Must there be an element $a\in G$ such that $|a^n|\ge c\cdot n$ for some $c>0$? My ...
2
votes
1answer
138 views

Reference request: flat surfaces

When writing a paper, I feel like to point out exact references to the following seemly easy facts concerning flat structures on a closed surface $\Sigma$ with negative Euler characteristic: The ...
1
vote
1answer
63 views

4D Duoprisms based on nonconvex polygons

A duoprism is a polytope that can be expressed as the Cartesian product of two polytopes (each of dimension $\ge 2$). Four-dimensional duoprisms in particular have been studied: $$P \times Q = \{ ...
19
votes
2answers
867 views

Tiling of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes, that is, it is a curve of constant width. (The width between two parallel tangents to the curve are independent of the ...
13
votes
2answers
249 views

Can any simplex shadow-project to a regular simplex?

Every triangle $A$ can be oriented in $\mathbb{R}^3$ so that its orthogonal projection (shadow) onto the $xy$-plane is an equilateral triangle $Q$:               ...
0
votes
1answer
52 views

Existence of shortest paths in complete Alexandrov spaces

Let $X$ be complete finite dimensional Alexandrov space with curvature bounded from below. Is it true that any two points can be connected by a shortest path? If this is not true in general, it it ...
0
votes
1answer
162 views

Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
1
vote
0answers
127 views

Products between metrics in a product of manifolds

In the "Einstein Manifold" book written by Arthur Besse, chapter 16, there is a notation of a manifold composed by the Cartesian product between two others: $(M_1\times M_2, f^p(g_1 \times g_2))$ ...
15
votes
2answers
678 views

Does equality of Laplacians imply Kähler?

This question follows on from this one. Let $(X, \omega)$ be a Hermitian manifold and define the Laplacians $\Delta_{\partial} = \partial\partial^* + \partial^*\partial$ and $\Delta_{\bar{\partial}} ...
4
votes
2answers
329 views

Terminology for metrics?

For some reason, I'm currently interested in the following relation - let $d,\delta$ be two metrics on some space $X$. We call the metrics _______ if there are some constants $C,E>0$ such that for ...
2
votes
1answer
116 views

Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
15
votes
5answers
736 views

Thales' semicircle theorem in higher dimensions

Thales semicircle theorem says that an angle inscribed in a semicircle is a right angle. Q1. Does a cone with apex on a hemisphere and encompassing the circular base have a solid angle ...
3
votes
0answers
55 views

On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces

Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm $$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m ...
3
votes
2answers
474 views

What's the name of this geometric mathematical modeling problem?

There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner. I know that this is a famous problem, but what is it called? ...
0
votes
0answers
69 views

Largest ball with fixed center in a a convex region

Let $x_0$ be a point contained inside a compact, convex set $C\subset\mathbb{R}^d$, which is of the form $C=\{x:f(x)\leq0\}$ for some explicit convex function $f$. Is there a computationally ...
2
votes
1answer
108 views

Shortest paths in Alexandrov spaces

Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact). Question 1. Is it true that every point of $X$ has a ...
0
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0answers
103 views

How large can a set of nearly equidistant points be?

Suppose that $D$ is a set of points in $\mathbb{R}^{k}$ such that all pairwise distances between them belong to $[1,1+\epsilon]$. It seems that such a set cannot be very large and that its ...