Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.
5
votes
1answer
344 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, ...
3
votes
0answers
54 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
1answer
142 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 ...
6
votes
2answers
187 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 ...
5
votes
2answers
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
3answers
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
1answer
109 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
0answers
72 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
$$
\alpha \,\left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right) \leq
...
1
vote
0answers
26 views
Euclidean embedding of a graph based on 1-ring neighborhood distances only
Consider a graph $(V,E)$, $\vert V \vert = n$ and weights $\{l_{ij}\}$, where $l_{ij}>0$ iff there is an edge connecting vertices $v_i$ and $v_j$. Distances beyond the 1-ring neighborhood are not ...
0
votes
0answers
32 views
Buseman function is regular on manifolds without boundary containing a line?
For an n-dim noncompact manifold M without boundary. Assume M contains a line $\gamma$. For every point $p \in M$, let $\widetilde{p\gamma(t)}$ be the geodesic from p to $\gamma(t)$.
Choose a ...
17
votes
2answers
315 views
An equivalence relation for norms
Let us say that two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a real vector space $V$ are strongly equivalent if there exists a constant $\lambda \geq 1$ such that
$$
\frac{1}{\lambda} \left( \|x\|_1 ...
3
votes
0answers
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
0answers
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:
...
0
votes
0answers
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
1answer
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 ...
0
votes
0answers
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 ...
6
votes
2answers
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 ...
1
vote
0answers
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 ...
8
votes
0answers
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 ...
9
votes
3answers
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 ...
5
votes
2answers
286 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 ...
0
votes
0answers
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 ...
7
votes
0answers
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 ...
6
votes
1answer
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 ...
14
votes
2answers
448 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
0answers
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 $2A$, ...
3
votes
1answer
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 ...
6
votes
2answers
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 $r,s$ ...
2
votes
0answers
44 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 ...
20
votes
3answers
720 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 ...
4
votes
1answer
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 ...
7
votes
2answers
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
3answers
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 ...
11
votes
2answers
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 ...
7
votes
0answers
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 ...
3
votes
1answer
182 views
A question on differential forms and integral invariants
The following question comes up in the study of metrics with the same unparameterized geodesics in Riemannian and Finsler geometry:
Question. Let $M$ be a closed manifold of dimension $2n+1$ and let ...
8
votes
4answers
291 views
More than $n$ approximately orthonormal vectors in $R^n$
This question was asked at math.stackexchange, where it got several upvotes but no answers.
It is impossible to find $n+1$ mutually orthonormal vectors in $R^n$.
However, it is well established ...
4
votes
2answers
131 views
Bound on Minimal Length of Vectors in Lattice and its Dual Lattice
Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
3
votes
0answers
54 views
3D objects with projections of constant area
Objects of constant width are well-known, and I have in my hand a (non-spherical) solid of constant diameter. The question arises:
Are there any non-spherical 3-dimensional objects such that their ...
4
votes
1answer
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 ...
7
votes
0answers
193 views
Surfaces with many (but not solely) closed geodesics?
Let $S$ be a closed surface embedded in $\mathbb{R}^3$,
let's say of genus zero.
I seek examples of $S$ with the following property:
If one selects a random any point $p$ on $S$, and a random
...
11
votes
0answers
191 views
Unit ball of smallest volume in a Hilbert geometry
In a letter to Felix Klein published in Mathematische Annalen 1895 (see here), Hilbert generalized the Cayley-Klein model of hyperbolic geometry by defining a metric on the interior of a convex body ...
0
votes
1answer
75 views
Constructing measures with support in a given set
I've recently come across the Frostmann Lemma (http://en.wikipedia.org/wiki/Frostman_lemma). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm ...
1
vote
2answers
69 views
Extreme points and centroid
It is well-known that the centroid of a triangle is the intersection point of its three medians. The medians happen to be area bisectors, but it seems that most (all?) other lines through the ...
0
votes
1answer
74 views
harmonic maps from cone to $S^2$ locally lipschitz?
Are the harmonic maps from a 2-dimensional cone to $S^2$ locally lipschitz or Holder continuous?
1
vote
2answers
57 views
Sampling uniformly from all possible line segments of a given length that fit inside a container
Consider that task of randomly placing a line segment of some length $L$ near a plane s.t. a point $p$ at the center of the line segment is at most a distance $H$ from the plane and intersections ...
12
votes
3answers
355 views
Characterization of discs
Let $D$ be a bounded simply connected region (open subset homeomorphic to the disc)
in the plane, containing the origin.
Suppose that for every line $L$ through the origin the intersection ...
5
votes
1answer
118 views
Maximal regions with given diameter
Let us call a bounded region $D$ in the plane maximal if the conditions $D\subset D'$ and
$\mathrm{diam} D'=\mathrm{diam} D$ imply $D'=D$.
Is it possible to describe all maximal regions?
The only ...
2
votes
1answer
36 views
Seeking criteria for “threadable” pairs of centrosymmetric polyhedra
Let $A$ and $B$ be origin-centered centrosymmetric polyhedra in $\mathbb{R}^3$:
"for every point $(x, y, z)$ [...] there is an indistinguishable point $(-x, -y, -z)$."
Say that $A$ and $B$ are ...
2
votes
0answers
44 views
Measure of points with small neighborhood in convex bodies
Let $K \subseteq \mathbb{R}^n$ be a "fat" convex body, i.e. one that contains a ball of radius 1. I'm interested in the following question about points $y \in K$: If you take a normally distributed $e ...
