# Questions tagged [mg.metric-geometry]

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

3,095
questions

**4**

votes

**1**answer

33 views

### Geodesic line with endpoints in interior of Riemannian manifold/ Alexandrov space

Let $X$ be a finite dimensional Alexandrov space with curvature bounded below and non-empty boundary. Let $\gamma$ be a shortest geodesic path in $X$ whose endpoints belong to the interior of $X$.
...

**3**

votes

**1**answer

95 views

### Tiling with similar tiles

Question 1: Is there a polygon $P$ that
cannot tile the plane
and
tiles the plane when copies of $P$ and some other polygon(s) all similar in shape to $P$ but of different size(s) can be used?
...

**16**

votes

**1**answer

587 views

### Can a fixed finite-length straightedge and finite-size compass still construct all constructible points in the plane?

I am hoping that the brilliant MathOverflow geometers can help me out.
Question 1. Suppose that I have a fixed finite-length straightedge and fixed finite-size compass. Can I still construct all ...

**1**

vote

**0**answers

47 views

### How can construct the equilateral $A''B''C''$ such that area of $A''B''C''$ is biggest

Let $ABC$ be arbitrary triangle in a plane. Let $A'B'C'$ and $A''B''C''$ be two equilateral triangles such that $A \in B'C'$, $B \in C'A'$, $C \in A'B'$ and $A \in B''C''$, $B \in C''A''$, $C \in A''B'...

**0**

votes

**0**answers

38 views

### Computer vision: retrieve rotation and translation matrix for stereo cameras [closed]

Summary
I have a list of corresponding image points from the left and right cameras of a stereo camera system. Setting the left camera as the origin, I want to find the extrinsic camera parameters (...

**6**

votes

**1**answer

325 views

### Who was Bickart?

The term "Bickart points" is often used for the foci of the Steiner circumellipse of a triangle. Who was Bickart, and what was the first publication to use the term?

**4**

votes

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302 views

+50

### Combinatorial optimization problem on the sums of differences of real numbers

We are given an increasing sequence $S_n$ of $n$ distinct real numbers $x_1, x_2, \ldots, x_n$, with $x_1=0$ and $x_n=1$. We define the complementary distance $D(q,p):=1-d(q,p)$, where $d(q,p):=|x_q-...

**3**

votes

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69 views

### Ultralimit of metric spaces vs. inductive limits of underlying topological spaces

Let $\{(X_n,d_n)\}_{n =1}^{\infty}$ be a sequence of bounded metric spaces such that:
$X_n \subseteq X_{n+1}$ is a metric subspace of $X_n$.
Let $\omega$ denote a non-principal ultrafilter (i.e.: ...

**13**

votes

**2**answers

1k views

### How many squares can be formed by using n points?

How many squares can be formed by using n points on a 3 dimensional space?
Like using 4 points, there is 1 square be formed
Using 5 points, still 1 square
Using 6 points, 3 squares can be formed

**3**

votes

**1**answer

50 views

### $AC^p$ curves and pointwise metric speed in abstract metric spaces?

For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if
$$
d(\gamma(s)...

**1**

vote

**0**answers

50 views

### n-dimensional polyhedron with special properties

I'd like to know if there exists a convex face transitive n-dimensional polyhedron with all dihedral angles equal to $\frac{2\pi}{3}$.
For n = 2,3,4 an example can be a regular hexagon, a rhombic ...

**2**

votes

**0**answers

89 views

### How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle

How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle?
See also:
Malfatti circles

**0**

votes

**1**answer

50 views

### Fast way to generate random points in 2D according to a density function

I'm looking for a fast way to generate random points in 2D according to a given 2D density function.
For instance something like this:
Right now I'm using a modified version of "Poisson disc&...

**1**

vote

**0**answers

42 views

### Maximizing distance between points on the positive surface of the unit hyper-sphere

Suppose we want to place $k$ ($k \geq 3$) points on the positive surface of a unit hyper-sphere in $\mathbb{R}^n$ ($n \geq 3$), where all coordinates of a point are positive, such that the minimum ...

**3**

votes

**0**answers

86 views

### Stewart's formula in plane geometry

In his book "Plane geometry and its groups", H. Guggenheimer proves the Stewart's formula:
If $A$, $B$, and $C$ are collinear, then for any point $Ρ$ in the plane
$$
PA^2 BC + PB^2 CA + ...

**22**

votes

**0**answers

585 views

### Curves on potatoes

On twitter recently, Robin Houston brought up this problem from a mathematical puzzle book of Peter Winkler:
The puzzle is attributed to the book "The mathemagician and the pied piper", and ...

**0**

votes

**0**answers

41 views

### Gromov-Hausdorff limit of compact surfaces with same boundary of equal areas

Define $$ A = (0,0,0), \ B=\bigg(\frac{1}{n},0,0\bigg), \ C=
\bigg(\frac{1}{n},\frac{1}{n},0\bigg) ,\ D= \bigg(
0,\frac{1}{n},0\bigg),\ E=
\bigg(\frac{1}{2n},\frac{1}{2n}, \frac{a}{n}\bigg)
$$ for ...

**10**

votes

**5**answers

681 views

### Examples of metric spaces with measurable midpoints

Given a (separable complete) metric space $X=(X,d)$, let us say $X$ has the measurable (resp. continuous) midpoint property if there exists a measurable (resp. continuous) mapping $m:X \times X \to X$ ...

**5**

votes

**0**answers

92 views

### Conv A = Dual B

I have two cones $A$ and $B$ in a Euclidean space.
I want to show that $\mathrm{Conv}\,A=\mathrm{Dual}\,B$; that is, $a\in\mathrm{Conv}\,A$ if and only if $\langle a,b\rangle\ge 0$ for any $b\in B$.
...

**4**

votes

**0**answers

80 views

### A lattice with Monster group symmetries

The book Mathematical Evolutions contains the following excerpt:
A last, famous, example is the following. It is known that in the space
of one hundred and ninety six thousand eight hundred and ...

**2**

votes

**1**answer

49 views

### Tangent cone of metric graph

I am starting to study some lecture notes about metric geometry and I would appreciate it if someone could some questions regarding the notion of the tangent cone.
Consider 3 half lines joined by ...

**1**

vote

**0**answers

35 views

### How dense can a transitive sets of points be?

How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points?
As a measure for density I use ...

**2**

votes

**0**answers

47 views

### Getting more out of Minkowski's convex body theorem in the case of non-convex bodies

Problem. In number theory one generally proves the finiteness of the Picard group of a number field using Minkowski's convex body theorem. The actual body $S_p$ of interest in the proof, depending on ...

**1**

vote

**0**answers

29 views

### Can an orderless set of inner product between N vectors determine unique structure of the vectors?

Suppose we have n vectors {a1,a2,a3,...,an} such that the sum of them is zero vector
a1+a2+a3+...+an=0
Now, we compute the inner product of each two vectors of them, i.e. we compute the Gramian matrix ...

**1**

vote

**1**answer

32 views

### How to define and compute the degree of congruence of two rigid polyhedra in same type with knowing vertex coordinates?

If I have two sets of points in 3-dimensional space, each sets of points are the coordinates of vertices of a polyhedron. The two polyhedra have same type, so we don't need to consider the topological ...

**2**

votes

**0**answers

75 views

### Generalization of Tucker circle, Conway circle and van Lamoen circle

Theorem 9.1 in this paper as follows is a generalization of Turker circle. Turker circles is a generalization of many circles as: Cosine Circle, circum circle, First Lemoine Circle, Gallatly Circle, ...

**8**

votes

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575 views

### A new theorem (discovered in 2013) equivalent to Brianchon theorem (the old theorem) discovered in XIX century?

In 2013, I found a new problem as follows: Let six points $A_1$, $A_2$, ...$A_6$ lie on a circle $(O_1)$, and the six points $B_1$, $B_2$,...,$B_6$ lie on another circle $(O_2)$. If the quadruples $...

**3**

votes

**1**answer

131 views

### Monotile that tiles when you apply a rubber band

My (non-mathematician) friend asked me a physics/tilings question that maybe someone here is interested in dissecting, or can point to the literature if this problem has been studied.
Does there ...

**1**

vote

**1**answer

68 views

### When is a $k$-distance-transitive graph already distance-transitive?

Call a (finite and connected) graph $k$-distance-transitive if its symmetry group acts transitively on the pairs in each one of the sets
$$D_\delta:=\{(i,j)\in V\times V\mid \mathrm d(i,j)=\delta\},\...

**4**

votes

**2**answers

74 views

### From a point and continuing reflection in $2n+1$ points then midpoint of the end point and the first point is fixed

Given $2n+1$ fixed points: $A_1, A_2,....,A_{2n+1}$ and point $P$. Let $B_1$ is the reflection of $P$ in $A_1$, $B_2$ is the reflection of $B_1$ in $A_2$,...., $B_{2n+1}$ is the reflection of $B_{2n}$ ...

**6**

votes

**1**answer

215 views

### Like Bradley’s conjecture (Four incenters lie on a circle) [closed]

Please don't close this question. Because there is simple configuration with 57 vote up, and don't close. Why you vote up that question and You vote to close this question?
A problem I posed at here ...

**4**

votes

**0**answers

84 views

### Can we combine the symmetries of two polytopes to create a more symmetric polytope?

Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$.
The symmetry group $\mathrm{Aut}(P_i)\subset\...

**1**

vote

**0**answers

44 views

### Euclidean embedding of the mesh

$M$ is a topological mesh, i.e. triple $M=(V,E,F)$, where $V$ is the vertex, $E$ is the edge and $F$ is the face, such that $M$ is homeomorphic to the sphere.
Suppose that we have a metric $l :E\...

**4**

votes

**0**answers

96 views

### Explicit homeomorphism between $L^p$ and Sobolev Space

From the Anderson-Kadec theorem, we know that all separable infinite-dimensional Banach spaces are homeomorphic. I'm wondering, is there an explicit such homeomorphism between $W^{p,k}(\mathbb{R}^n)$ ...

**2**

votes

**1**answer

96 views

### 3D similarities and quaternions?

As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form
$$\forall z \...

**2**

votes

**1**answer

50 views

### Algorithms for finding minimal ball covers

My question is whether there exist algorithms to determine a minimal (or close to minimal) cover of a Riemannian manifold (or some subset thereof) with balls of a fixed radius r>0.

**5**

votes

**1**answer

81 views

### A polytope with congruent facets and an insphere that is not facet-transitive?

Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties?
All facets are congruent,
it has an ...

**4**

votes

**0**answers

60 views

### Closed curves with minimal total curvature in the unit circle

Chakerian proved in this paper that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L.
In this later paper Chakerian gave a simpler proof and noted that ...

**7**

votes

**2**answers

220 views

### Is a polytope that has in-spheres for faces of all dimensions already regular?

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points).
A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent. So a 0-in-sphere ...

**1**

vote

**0**answers

72 views

### Lattice points in hypercubes

Let $ (\Lambda_n) $ be a family of lattices, $ \Lambda_n \subset \mathbb{Z}^n $, with $ \det\Lambda_n \sim n $ as $ n \to \infty $ (meaning $ \lim_{n\to\infty} n^{-1} \det\Lambda_n = 1$). I am ...

**4**

votes

**2**answers

260 views

### Is the intersection of two distinct sufficiently small metric spheres always empty, a point or a metric sphere of lower dimension?

Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-...

**3**

votes

**1**answer

41 views

### Tilings of lattice polytopes by transformations of lattice polytopes

A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...

**9**

votes

**2**answers

227 views

### Generalized figures of constant width

Is it known which plane figures $Q$ can rotate touching three given circles $A$, $B$, and $C$?
This question was asked by Lazar Lyusternik in 1946, there is only one reference to this paper that ...

**1**

vote

**0**answers

95 views

### On some optimal containers of a set of points on the 2D plane

Given a set of N points in general position on the plane, the problem is to give efficient algorithms to find
the smallest semicircular region (semidisk) that contains the points
the smallest ...

**2**

votes

**0**answers

127 views

### What is the smallest dimension that allows finding $n$ points at distances $|x_i-x_j|^{\delta/2}$, where $0<\delta<1$, and $x \in \mathbb{R}^n$?

Let $x_1,\cdots,x_n \in \mathbb{R}$, are there $\xi_1,\cdots,\xi_n \in \mathbb{R}^s$, such that
$|x_i-x_j|^{\delta}=||\xi_i-\xi_j||^2$, $0<\delta<1$, what is the smallest $s$ to guarantee the ...

**7**

votes

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119 views

### Hanging a cube with string

This is a variation on a (much) earlier MO question, Hanging a ball with string.
Here instead the task is to arrange a net of string to hang
a unit cube. Assume:
The string is inelastic.
There is no ...

**2**

votes

**0**answers

49 views

### Finding a superbase in a lattice of Voronoi first kind

An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that
$\{b_1,\ldots,b_n \}$ is a ...

**5**

votes

**1**answer

103 views

### What is known about the duals of cyclic polytopes?

What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)?
In even dimensions, all facets of the dual are ...

**4**

votes

**1**answer

185 views

### Number of points in a lattice and an oblong box

I have a very simple question in geometry of numbers. (It is a slight modification of Counting points on the intersection of a box and a lattice .) There's a bound I can easily prove, and it's good ...

**1**

vote

**1**answer

46 views

### tetrahedral interpolation and integration along a segment

Let's say we have a several tetrahedrons $T_i$ whose faces touch so that each face belong to two tetrahedrons. Each tetrahedron contain a value $V_{i}$.
Given a position $P$ inside the tetrahedron $...