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

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12
votes
1answer
297 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 ...
14
votes
0answers
427 views

An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
9
votes
0answers
159 views

Multiplicity of ball covering

Background. My questions are motivated by the following: A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete and Computational Geometry, 8 (1992) 109-130) considered the ...
12
votes
2answers
582 views

Right triangle with edge lengths equal to regular unit polygon edge lengths

This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...
3
votes
1answer
143 views

Restricted isometry

Here is a problem that has been bugging me for a while. Let $\| \|$ be a norm over $\mathbb{R}^n$, let $C$ be a convex subset of $\mathbb{R}^n$ with non-empty interior, and let $f: (C,\|\|) ...
1
vote
0answers
90 views

Entangled helical knots

Consider a pair of disjoint, congruent helices $H_1$ and $H_2$ passing through one another in the following sense. (Caveat lector: This question is not of general interest! It is also long.) $H_1$ is ...
11
votes
1answer
389 views

Heronian triangle with two sides that are prime

Can any prime number form a Heronian triangle with a second prime as another side? I cannot find a second prime to form a Heronian triangle with either 23 or 167. I have checked up to the 10^7th prime ...
1
vote
1answer
164 views

Probabilistic Johnson-Lindenstrauss Lemma for arbitrary points

Consider the following standard formulation of the Johnson-Lindenstrauss lemma: Lemma (JL). For any $0<\epsilon < 1$ and any integer $n$, let $k$ be a positive integer such that $k\geq ...
6
votes
0answers
171 views

Reversing shortest paths among unit disks

Twas the night before Christmas, and throughout M.O. Not a question was posted, not even by Joe. Well, let me remedy that. :-) Let the plane contain a number of ...
3
votes
2answers
185 views

Equipartition of the circle [closed]

Browsing an old technical studies pupil's school book, I have found the description of a method to place at equal distance $N$ points on the circumference of a circle. I am looking for a proof of this ...
3
votes
1answer
160 views

Closed surface with uncountably many conic points?

Let $M^2$ be a closed surface, say the 2-sphere. Is there any example of metric on it such that there are uncountably many points are conic and the metric is smooth elsewhere? We call $p\in M$ a ...
-2
votes
1answer
159 views

Equivalence relations on powerset of R^2

Let A and B be two subsets of R^2. I define the relation T(A,B) to hold between A and B iff there exists a translation f on R^2 such that the image set of A under f is B. It is easy to prove that T is ...
5
votes
1answer
157 views

Shortest curve with given convex hull

Suppose $S\subset\mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to S$ is a continuous curve that passes through every extreme point of $S$, i.e., the convex hull of $\Gamma([0,1])$ is $S$. ...
3
votes
0answers
89 views

Lattices achieving best density

Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ ...
18
votes
0answers
331 views

The Eyeball Theorem generalized

I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher ...
10
votes
1answer
640 views

What is the longest algebraic curve?

Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$. Is $L(d)\leq d P(E)/2$, where $P(E)$ is the ...
0
votes
1answer
136 views

Lipschitz boundary vs rectifiable curve boundary

I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this ...
13
votes
3answers
659 views

Are infinite planar graphs still 4-colorable?

Imagine you have a finite number of "sites" $S$ in the positive quadrant of the integer lattice $\mathbb{Z}^2$, and from each site $s \in S$, one connects $s$ to every lattice point to which it has a ...
2
votes
2answers
109 views

Nonplanar equilateral lattice “pentagons”

It is well-known that no two-dimensional point lattice contains a regular pentagon. (See for example http://mathworld.wolfram.com/LatticePolygon.html.) The same is true for lattices in $\mathbb{R}^n$, ...
34
votes
3answers
1k views

What fraction of the integer lattice can be seen from the origin?

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes ...
3
votes
2answers
197 views

Place N points in a 3d cube in a way that maximizes the minimum of their pairwise distances

Place $N$ points in a 3d cube in a way that maximizes the minimum of their pairwise distances. The problem can easily be solved for $N\lt5$, but how to proceed for larger $N$?
5
votes
1answer
175 views

Which lenses can be squared?

A lune and a lens are both planar figures delineated by two circular arcs; the difference is that a lune has a concave and a convex arc (it is one circle minus another) whereas a lens has two convex ...
4
votes
1answer
141 views

Bounding the perimeter of a geodesic triangle in spaces of non-positive curvature

This is probably an easy question, but I don't know any Riemannian geometry and a literature search hasn't helped. Any help (e.g. providing a reference) would be greatly appreciated. For a triangle ...
1
vote
1answer
33 views

Length of the transversal for surfaces with cusps

In Peter Buser's Geometry and Spectra of Compact Riemann Surfaces he shows that the length of the transverse curve to a geodesic in a pants decomposition on a compact hyperbolic surface has length a ...
3
votes
1answer
195 views

Map from a convex polygon that increases distance

At the risk of asking an extremely stupid question, suppose that $P\subset\mathbb{R}^2$ is a convex polygon with area $1$ that contains the origin, and let $r$ denote the farthest distance between the ...
4
votes
3answers
281 views

On discrete version of curve shortening flow

One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the ...
16
votes
2answers
407 views

What kind of probability distribution maximizes the average distance between two points?

If $f$ is a probability distribution on the unit disk in $\mathbb{R}^2$, and $X_1$ and $X_2$ are two independent samples from $f$, then what is the distribution $f^*$ that maximizes the average ...
9
votes
0answers
222 views

Periodic orbits of a spinning ball in a square

Periodic orbits of a billiard ball bouncing in a square have been well-studied. I am seeking similar analysis of what is sometimes called a rough ball, one whose high friction causes it to pick up ...
6
votes
1answer
101 views

Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$. It is clear that there is a constant ...
4
votes
1answer
74 views

Polyhedra with minimal edge length

Given a fixed volume and fixed surface area I would like to construct polyhedra that minimize the total length of the edges. This seems like a straight-forward problem to solve by brute force for ...
6
votes
0answers
167 views

Equiareal shapes in $\mathbb{R}^d$

There was quite a bit of work on the so-called equichordal problem throughout the 20th century, to decide if some plane convex curve could have two equichordal points. A point is equichordal for a ...
5
votes
1answer
225 views

Regular lattice polygons

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within ...
4
votes
1answer
245 views

Inequality involving the side lengths of a quadrilateral

If $a$, $b$, $c$ and $d$ are the four sides of a quadrilateral, the problem is to show that $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. I've verified it to be true for quite a large number of ...
2
votes
0answers
93 views

Definition of self-dual polytope

Given a d-polytope $P$ we define the c-dual polytope as $P^\ast = \{y\in R^d \mid x\cdot y\geq -c, \forall x\in P\}$. Then I say that a polytope is c-polar self-dual if $P=P^\ast$. I cannot find this ...
1
vote
0answers
52 views

Finding special vectors generated by a matrix

Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix. Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...
1
vote
0answers
44 views

Covering a set of points by bounded geometric object/objects

1) Let $S$ be a set of $n$ points in $R^d$. Now, given a bounded geometric object $G$, the problem is to check whether $S$ can be contained in $G$. 2) Also, in general setting, the problem is to ...
2
votes
1answer
105 views

Is there interesting structure in the space of Voronoi diagrams?

Given a finite set $X = \{x_1,\dots,x_N\}$ of points in $\mathbb{R}^n$, let $V_j(X)$ be the interior of the Voronoi polytope corresponding to $x_j$, i.e., the interior of the set of points closer to ...
4
votes
0answers
112 views

construction of heat kernels for non-compact manifolds with boundary

Recently, I am studying heat semigroup for noncompact manifolds with boundary. In Issac Chavel's book "eigenvalues in Riemannian geometry". "Given a noncompact Riemannian manifold, it need not be ...
5
votes
0answers
164 views

4D polytope analogues of the icosahedron/Rogers-Ramanujan continued fraction relationship?

The formula for the j-function which employs polynomial invariants of the icosahedron, $$j(\tau)=-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5}$$ where, ...
0
votes
0answers
46 views

Estimating the mean of a bivariate distribution by randomly sampling variates and rounding them to integer coordinates

Imagine that we have a particle sampling positions on a two-dimensional plane according to a bivariate probability distribution: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, ...
1
vote
2answers
253 views

Preservation of injectivity radius

Suppose I have a smooth, noncompact manifold $M$ with metrics $g_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that $$ C^{-1} d_1(x,y) \leq d_2(x,y) \leq C d_1(x,y)$$ where $d_i$ are the ...
10
votes
1answer
301 views

Tverberg's theorem in CAT(0) spaces

Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$: Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - ...
6
votes
0answers
157 views

Choquet theory and Hilbert's fourth problem

The following text is an attempt to see Hilbert's fourth problem in a new light. Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...
1
vote
0answers
157 views

Sampling a two-dimensional Gaussian distribution at points along an integer lattice

Please consider a two-dimensional Gaussian of the general form: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, where $C$ is the peak of the Gaussian, i.e. the point at which the ...
2
votes
1answer
193 views

Optimal, conformal diffeomorphisms between two surfaces in 3D

Let $S_1$ and $S_2$ be two smooth, closed surfaces embedded in $\mathbb{R}^3$. Q. Is there a natural definition of the optimal, conformal diffeomorphism between $S_1$ and $S_2$? I am imagining ...
7
votes
1answer
303 views

Shrink polygon to a specific area by offsetting

I have a 2D polygon that I want to shrink by a specific offset (A) to match a certain area ratio (R) of the original polygon. Is there a formula or algorithm for such a problem? I am interested in a ...
2
votes
0answers
143 views

Comparing two metrics on the space of infinite sequences and relating open and closed sets

Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it $$ d(u,v) := \frac{1}{2^r} \mbox{ with } ...
2
votes
1answer
119 views

Circumscribed ellipsoid of minimum Hilbert-Schmidt norm

Let $K\subseteq \mathbb{R}^n$ be a full-dumensional convex body. The Löwner ellipsoid of $K$ is the unique ellipsoid of smallest volume containing $K$. My question is about a related object: the ...
2
votes
1answer
160 views

About the inscribed sphere and the exspheres of a $n$-dimensional simplex

Let us consider $n$-dimensional simplex $K$ in $n$-dimensional Euclidean space. Let $r_0$ be the radius of the inscribed sphere of $K$, and let be $r_1, r_2, \cdots, r_{n+1}$ be each radius of the ...
15
votes
2answers
519 views

“Derived” polyhedra and polytopes

The notion of derived polygon is natural and leads to remarkable convergence. Start with a polygon, and replace it by locating a point on every edge a fraction $\alpha$ between the two endpoints. For ...