**3**

votes

**0**answers

22 views

### Is there a sideways-walking rolling convex body?

Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$.
Place $K$ on an inclined plane, and let it roll down the plane,
under some reasonable assumptions of friction between $K$ and
the plane, ...

**2**

votes

**0**answers

53 views

### Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-)

I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the ...

**6**

votes

**1**answer

140 views

### Limit of distance between two random points in a unit-radius $n$-sphere

This is a companion contrast to the earlier analogous question for unit $n$-cubes,
where the answer (provided by several respondents) is $\infty$ .
What is the limit, as $n \to \infty$, of the ...

**5**

votes

**2**answers

185 views

### Limit of distance between two random points in a unit $n$-cube

What is the limit, as $n \to \infty$, of the expected distance between two
points chosen uniformly at random within a unit edge-length hypercube
in $\mathbb{R}^n$?
For $n=1$, the average ...

**1**

vote

**5**answers

1k views

### Quadrilateral from 4 random points

Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points?
I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q.
But ...

**5**

votes

**2**answers

327 views

### Geometric explanation of Hutton's formula?

$$\frac{\pi}{4} = 2 \tan^{-1} \frac{1}{3} + \tan^{-1} \frac{1}{7} \;.$$
Is there some geometric construction that explains this beautiful equation
(known as Hutton's formula)?
Perhaps a "proof without ...

**3**

votes

**1**answer

259 views

### Shortest geodesic loop vs. shortest periodic geodesic

Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic?
For example, is this true for small ...

**3**

votes

**3**answers

221 views

### Points contained in a disk [on hold]

I have a question, but not sure how to prove this.
We are given $n$ points in the Euclidean plane such that there exists no disk of radius $a$ which contains all of the points.
Conjecture: There ...

**8**

votes

**1**answer

108 views

### Random walk on a Penrose tiling

Pólya proved that a random walk on $\mathbb{Z}^2$ almost surely returns to the
origin, or, equivalently, returns to the origin infinitely often.
It was subsequently established that in $\mathbb{Z}^3$, ...

**5**

votes

**0**answers

71 views

### Geometry of the metric cone

Let us say that two metrics $d$ and $d_0$ on a set $X$ are related if there exist positive constants $0 < \alpha \leq \beta$ such that
$$
\alpha \,\left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right) \leq
...

**1**

vote

**0**answers

25 views

### Euclidean embedding of a graph based on 1-ring neighborhood distances only

Consider a graph $(V,E)$, $\vert V \vert = n$ and weights $\{l_{ij}\}$, where $l_{ij}>0$ iff there is an edge connecting vertices $v_i$ and $v_j$. Distances beyond the 1-ring neighborhood are not ...

**17**

votes

**2**answers

314 views

### An equivalence relation for norms

Let us say that two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a real vector space $V$ are strongly equivalent if there exists a constant $\lambda \geq 1$ such that
$$
\frac{1}{\lambda} \left( \|x\|_1 ...

**41**

votes

**2**answers

933 views

### How many unit cylinders can touch a unit ball?

What is the maximum number $k$ of unit-radius cylinders with mutually disjoint interiors that can touch a unit ball?
By a cylinder I mean a set congruent to the Cartesian product of a line and a ...

**9**

votes

**1**answer

204 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 ...

**9**

votes

**3**answers

693 views

### Could a perfect squared square be split into two perfect squared squares?

This is a geometric puzzle though it might conceivably
also define a special class of Pythagorean triples.
A perfect squared square PSS is a square (as a plane figure)
partitioned into smaller ...

**0**

votes

**0**answers

32 views

### Buseman function is regular on manifolds without boundary containing a line?

For an n-dim noncompact manifold M without boundary. Assume M contains a line $\gamma$. For every point $p \in M$, let $\widetilde{p\gamma(t)}$ be the geodesic from p to $\gamma(t)$.
Choose a ...

**4**

votes

**7**answers

2k views

### Uniformly Sampling from Convex Polytopes

How to choose a point uniformly from a convex polytope $P \subset [0,1]^n$ defined by some inequalities, Ax < b ? (Here A is an m-by-n matrix, x is n-by-1 and b is m-by-1.) I imagine that you ...

**3**

votes

**0**answers

135 views

### Is there such a matrix in $SO(n)$?

Given two $n$ dimensional positive definite matrices $A', B'$, is there a matrix $O \in SO(n)$ such that $A=O A', B=O B'$ and
$$
\frac{A_{ij}}{\sqrt{A_{ii}A_{jj}}} = ...

**4**

votes

**0**answers

64 views

### Regular cross-sections of a dodecahedron; analogous sections of 4-polytopes

One can intersect a dodecahedron with a plane and
obtain an equilateral triangle, a square, a regular pentagon,
a regular hexagon, and a regular decagon:
...

**7**

votes

**0**answers

181 views

### quantitative version of the rigidity of the 2-sphere

I am looking for a quantitaive version of the following theorem:
A compact surface with $K\equiv 1$ is isometric to the round sphere.
Of course I get the Berger, Brendle-Schoen Theorem which insures ...

**0**

votes

**0**answers

63 views

### Isometry groups acting transitively [migrated]

Let $X$ be a metric space and $G$ be its group of isometries.
1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...

**2**

votes

**1**answer

97 views

### Omit each vertex in turn of convex polygon: Iterative limit?

Let $P=P_0$ be a convex polygon of $n$ vertices $v_k$.
Let $P_{i+1}$ be the convex polygon obtained by intersecting the halfplanes
determined by the lines through every other vertex.
Below, $P_0$ is ...

**6**

votes

**1**answer

133 views

### Asymmetric metrics and cohomology

If $(X,d)$ is a metric space and $f : X \rightarrow \mathbb{R}$ is a Lipschitz function with Lipschitz constant $k < 1$, then the function
$$
D(x,y) := d(x,y) + f(y) - f(x)
$$
defines an asymmetric ...

**6**

votes

**2**answers

137 views

### get a point in polygon (maximize the distance from borders)

I have several 2D polygons represented by lists of xy-coordinates of their vertices.
It is needed to get several points inside the polygon so that they lie possibly far from the polygon's borders ...

**0**

votes

**0**answers

46 views

### Guassian upper bound of the heat kernel implies ultracontrativity?

For spaces satisfies uniformlly local doubling and Poincare inequality (for example, Riemannian manifold with Ricci curvature bounded below RCD(K,N)).
By Sturm's paper, we have bounds on the heat ...

**13**

votes

**4**answers

994 views

### Decidability of tiling R^2

Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane?
I know the ...

**8**

votes

**0**answers

132 views

### Characterization of Volumes of Lattice Cubes

Here is a problem that came up in a conversation with a professor after I made a false assumption about the geometry of $\mathbb{Z}^n$. I do not know if he knew the answer (and told me none of it) and ...

**1**

vote

**0**answers

49 views

### vertex representation and half-space representation of a polytope [migrated]

i'm not a mathematician. So, my question may be stupid. Hope that it won't make you angry.
Is there any mathematical relationship between the vertex representation and the half-space representation ...

**3**

votes

**1**answer

82 views

### Is any finite collection of points contained in a cut and project set with $\mathbb{R}^d$ as internal space?

In "Meyer Sets and their Duals" Moody proves that any Meyer set union a finite number of points is again a Meyer set. Additionally, any Meyer set is contained in a finite union of model sets whose ...

**4**

votes

**3**answers

278 views

### Brownian motion on Metric spaces

Is there a generalization of Brownian motion to general metric spaces (which should probably be length spaces)?
This should be a process satisfying
$$d(B_t, B_s) \sim \mathcal{N}(0, t-s)$$
and such ...

**5**

votes

**2**answers

284 views

### Volume-preserving projective transformations are isometries

What is a simple, elementary proof of the following result?
A continuously differentiable map from the unit sphere $S^n \subset \mathbb{R}^{n+1}$ $(n > 1)$ to itself that preserves volumes and ...

**7**

votes

**0**answers

78 views

### Longest induced cycles in random geometric graphs near criticality

We make a random geometric graph $X(n;r)$ as follows. Choose $n$ points uniformly, independently, in the unit square $[0,1]^2$, for vertices, and then connect a pair of vertices $\{ p,q \}$ by an edge ...

**9**

votes

**3**answers

630 views

### Neusis constructions

Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
See http://en.wikipedia.org/wiki/Constructible_number and ...

**0**

votes

**0**answers

88 views

### Generalized Sphere Kissing Problem

I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching ...

**3**

votes

**1**answer

163 views

### Separable Banach spaces which are absolute Lipschitz retracts

A subset $F$ of a metric space $M$ is called a Lipschitz retract of $M$ if there is a Lipschitz map from $M$ onto $F$ which coincides with the identity on $F$. A metric space which is a Lipschitz ...

**14**

votes

**2**answers

447 views

### Spearing rolling hula hoops

Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, ...

**2**

votes

**0**answers

47 views

### Intrinsic volumes of a family of convex sets $\{K_n\}$

Consider a family of convex sets $\{K_n\}$ such that $K_n \subset \mathbb{R}^n$ for each $n$. The kinds of sets one might be considering could be, for instance,
$K_n$ is the cube of side $2A$, ...

**6**

votes

**2**answers

149 views

### Intersecting Sets of Pythagorean Triples with Common Hypotenuses

For any $r\in\mathbb{N}$, let $A_r$ denote the set of all natural numbers that are potentially a side of a Pythagorean triple with hypotenuse $r$.
Given any $N\in\mathbb{N}$, does there exist $r,s$ ...

**20**

votes

**3**answers

719 views

### “Paradoxes” in $\mathbb{R}^n$

One may think of this question as a duplicate of this one. I see it more like an extension.
The "inscribed sphere paradox" discussed in the aforementioned question states that if you inscribe a ...

**2**

votes

**1**answer

103 views

### Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...

**2**

votes

**0**answers

43 views

### Smallest distribution of points with genuinely different clusterings

An hierarchical clustering algorithm for (finite) sets of points in a given metric space is essentially determined by its linkage criterion, which defines the distance between arbitrary (finite) sets ...

**4**

votes

**1**answer

330 views

### Triple bubble conjecture: Natural candidate?

Is there a standard natural candidate surface for
the shape that encloses three given volumes
in $\mathbb{R}^3$ and has minimal surface area?
I know the planar triple bubble conjecture was ...

**2**

votes

**1**answer

280 views

### Straight line on the Poincare disk hitting points almost everywhere

Consider the tiling of the Poincare disk $\mathbb{D}$ by identified octagons (i.e., representing a torus with genus 2). Suppose inside each octagon is a subset A such that the octagon minus A is a ...

**7**

votes

**2**answers

177 views

### How to find a tetrahedron that covers four points?

I’m looking for an explicit formula for the vertices of a regular tetrahedron that covers four given points. In particular:
Given four distinct real numbers $a_1$, $a_2$, $a_3$, $a_4$, is there a ...

**9**

votes

**3**answers

455 views

### Does the centroid depend continuously on the curve?

Let $\gamma$ be a piecewise smooth curve in $\mathbb{R}^n$. Recall that the centroid of $\gamma$ is the point $(\overline{x}, \overline{y})$ where $\overline{x}$ is the average value of $x$ on ...

**4**

votes

**0**answers

117 views

### Convexified threshold of a function

Upd. The question in a nutshell: find convex set on plane which is «closest» to a given non-convex set.
It is given integrable function $0\leq f(x,y)\leq 1$ with bounded support: $f(x,y)=0$ when ...

**4**

votes

**2**answers

294 views

### Sets of vectors related by a rotation

We have a two sets of vectors ($\mathbb{C}^d$), $A=\{ v_1, \ldots v_n\}$ and $B=\{u_1, \ldots u_n\}$.
The question is if there is an efficient solution (polynomial in $n$) for checking whether $A$ ...

**11**

votes

**2**answers

283 views

### Triangle with largest perimeter in a convex region

What is the largest value of $r$ such that the following statement is always true?
"Let $C$ be a convex region with area $1$. There must exist a triangle contained in $C$ whose perimeter is at least ...

**4**

votes

**1**answer

182 views

### Boundaries of relatively hyperbolic groups

When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...

**2**

votes

**1**answer

132 views

### Distance measure for noisy $SE(3)$ transforms

I have a transformation $T \in SE(3)$ parameterized by a mean quaternion $q$ with covariance matrix $\Sigma_q \in R^{4\times4}$ and a mean translation $t \in R^3$ with covariance matrix $\Sigma_t \in ...