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

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17
votes
2answers
489 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
146 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
152 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:             ...
2
votes
1answer
114 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
59 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 ...
8
votes
2answers
454 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 ...
18
votes
1answer
317 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
799 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
341 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
109 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
95 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 ...
15
votes
1answer
337 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
476 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}, ...
3
votes
0answers
79 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
238 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
157 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
59 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 ...
21
votes
3answers
875 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 ...
5
votes
1answer
405 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
208 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 ...
10
votes
4answers
539 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
404 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
204 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
199 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 ...
9
votes
4answers
442 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
159 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
61 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 ...
5
votes
1answer
248 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
211 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
300 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
85 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
88 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
76 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
102 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
371 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
130 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
40 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
56 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 ...
4
votes
1answer
505 views

Probability distribution or the distance between two points in $n$-dimensional Euclidean space after a random perturbation of one point

Take two points, $p_0$ and $p_k$, in $n$-dimensional Euclidean space, where $d(p_0,p_k)$ is the distance between the points. Now, draw an $n$-sphere of radius $r$ centered on $p_0$ and uniformly ...
11
votes
2answers
332 views

The intersection of a circle and a rank 3 subgroup of the plane

Let $A$ be a rank 3 subgroup of the Euclidean plane, i.e. $A = \mathbb{Z} v_1 + \mathbb{Z} v_2 + \mathbb{Z} v_3$, where $v_1, v_2, v_3 \in \mathbb{R}^2$ are three $\mathbb{Q}$-linearly independent ...
8
votes
1answer
479 views

Is this knot invariant already treated somewhere in the literature?

Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$ We can turn $Y_K$ into a metric space by considering the distance induced by ...
1
vote
0answers
53 views

Twisted calibrations and sufficient conditions on homology of sub-manifolds

I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...
3
votes
1answer
218 views

Computational approach deciding whether a set of Wang Tile could tile the space up to some size

As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...
5
votes
3answers
292 views

How hard is it to determine if a weighted graph can be isometrically embedded in R^3?

Consider a graph $G$ with nonnegative edge weights. Question: In $\mathbb{R}^3$, how hard is it to assign coordinates to vertices such that the Euclidean length of each edge is equal to its weight? ...
9
votes
1answer
287 views

A question about tiling Hilbert Space

Let H be an infinite dimensional and separable Hilbert Space. Let e be a positive real number-which can be arbitrarily small. Does there exist a denumerably infinite set S of pairwise disjoint and ...
3
votes
1answer
207 views

Planar linkage that traces a circle from its exterior?

Q. Is there a linkage in the plane that traces out a circle $C$ in such a manner that the interior of the disk bounded by $C$ is never intersected by any link througout the motion? What I ...
5
votes
1answer
239 views

Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?

In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?". Can someone explain what are the major ...
13
votes
0answers
406 views

Blocking light with mirrored convex objects

There is a long-unsolved problem posed by Janos Pach, sometimes known as the enchanted forest problem, which asks if it is possible to block a point light source in the plane from reaching infinity by ...
1
vote
2answers
170 views

Local vs distance function metric structures

The geodesic distance $d(p_1, p_2)$ on a geodesically complete, Riemanian manifold is a metric in the sense of a metric space metric. I'd like to know when other infinitesimal metric structures (e.g. ...
2
votes
1answer
168 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 ...