Questions tagged [mg.metric-geometry]

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

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10 votes
0 answers
727 views

Topological dimension, Hausdorff dimension, and Lipschitz mappings

I can prove the following result. Here $\operatorname{dim} X$ stands for the topological dimension and $\mathcal{H}^n$ denotes the Hausdorff measure. Theorem. Suppose that $f:\mathbb{R}^n\supset\...
5 votes
1 answer
415 views

Golden ratio as a property of conic section (is it known?)

I am looking for a proof of a discovery as follows: Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $(\...
2 votes
1 answer
215 views

Is the Jaccard distance between probability vectors a metric?

Let X and Y be probability vectors, meaning that X = $[x_1, x_2, ..., x_n]^T$, where $x_i\leq 1$ and $\sum_{i=1}^{n}x_i=1$ (Y is defined similarly). Define the Jaccard distance as \begin{equation} ...
7 votes
1 answer
189 views

Does there exist a countable metric space which is Lipschitz universal for all countable metric spaces?

Is there a countable metric space $U$ such that any countable metric space is bi-Lipschitz equivalent to a subset of $U$? How about $c_{00}(\mathbb{Q})$ where $\mathbb{Q}$ is the rational numbers? ...
3 votes
1 answer
562 views

Maximizing the distance sum of some points inside a circle

Consider $n$ points $\{p_i\}_{i=1}^n$ located inside or on a circle with radius $r$ in the plane. The question is: how to place the $n$ points so that the sum of inter-point distances, $$J=\sum_{i=1}^...
1 vote
1 answer
78 views

Equal products of triangle areas

Can you prove the following claim: Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its ...
1 vote
1 answer
340 views

Thirteen-point conic and four-point line, are they new?

We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and ...
6 votes
2 answers
617 views

How to define a Voronoi reduced basis?

Let $\Lambda$ be an $n$-dimensional lattice with basis $b_1,\ldots,b_n$. The problem of finding a "good" basis for $\Lambda$, or reducing a "bad" basis into a good one, is a very active area of ...
1 vote
1 answer
142 views

Generalizing Bottema's theorem

Can you provide another proof for the claim given below? Claim. In any triangle $\triangle ABC$ construct triangles $\triangle ACE$ and $\triangle BDC$ on sides $AC$ and $BC$ such that $\frac{AE}{AC}...
5 votes
0 answers
202 views

Covering the sphere with an approximately planar grid

Consider a triangulation of a radius $R$ sphere into $n$ triangles. Must $Ω(\sqrt n)$ triangles have $Ω(1)$ relative difference from being an equilateral triangle of area $4πR^2/n$?  ($Ω$ is from ...
1 vote
1 answer
141 views

On convex polygons contained in convex polygons

In what follows '$n$-gon' stands for '$n$-vertex polygonal region'. Question: Given a convex $n$-gon $C$, find the smallest convex region $R$ such that $C$ is the smallest $n$-gon that contains it. ...
2 votes
1 answer
206 views

What is the center of minimum distance of a region?

Suppose we have a compact plane region $R$ (not necessarily convex or connected). I am working in a problem which involves the point $p$ in $R$ that is, in average, the closest to every other point. ...
1 vote
0 answers
167 views

Does the Hausdorff dimension characterise CAT(0) spaces having some bilipschitz balls?

It is well-known that the Hausdorff dimension is invariant under bi-Lipschitz mappings. I would be interested in a specific converse of this invariance. Let $X$ and $Y$ be two CAT(0) spaces having the ...
3 votes
1 answer
92 views

Equal sums of line segments

I would like to see a proof of the following Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection ...
8 votes
2 answers
334 views

Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...
10 votes
2 answers
886 views

Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?

This is a cross-post. While working on a variational problem, I have reached to the following question. Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$...
10 votes
1 answer
2k views

Curves of constant curvature on an ellipsoid

It is not difficult to see that the curves of constant geodesic curvature on a geometric sphere are all circles: simple, closed curves that are geometric circles lying in a plane:    &...
3 votes
2 answers
146 views

Asymmetry of projections

A possible measure of asymmetry for a convex body $K \subset \mathbb{R}^n$ is the affine-invariant quantity $$ \alpha_n(K) := \frac{\textrm{vol}(K - K)}{2^n\textrm{vol}(K)} . $$ Indeed, the Brunn-...
1 vote
1 answer
203 views

Stability of isoperimetric inequality

Let $S$ be subset of $\mathbb{R}^n$ with perimeter 1. Isoperimetric inequality states that then the volume of $S$ is not greater than $V_n$, where $V_n$ is the volume of a ball in $\mathbb{R}^n$ with ...
4 votes
1 answer
169 views

Convergence of semi-concave functions on Alexandrov spaces

I will use the terminology of this paper by A. Petrunin: https://arxiv.org/pdf/1304.0292.pdf Let a sequence of $n$-dimensional Alexandrov spaces $\{X_i\}$ of curvature at least $-1$ converges to an ...
1 vote
0 answers
109 views

A generic question on circles associated with a triangle

This question is inspired by two posed by P.Terzić (both given elegant synthetic proofs by F. Petrov). The starting point is a triangle $ABC$ and a triangle centre $G_1$. There are two classical ...
7 votes
1 answer
267 views

Are two quasi-isometric, isomorphic on large enough balls, transitive graphs isomorphic?

Take two transitive graphs $X,Y$ (potentially directed and edge-labelled, e.g. Cayley graphs). Assume $X,Y$ are quasi-isometric with constant $K$, i.e. there exists a function $f:VX \to VY$ ($VX,\,VY$ ...
6 votes
0 answers
132 views

Nearby convex set in a nearby space

Let $K$ be a convex set in a CAT(0) space $X$. Suppose $X'$ is a CAT(0) space that is very close to $X$. Is there a convex set $K'\subset X'$ that is close to $K\subset X$? Two spaces $X$ and $X'$ ...
95 votes
11 answers
13k views

Is it possible to capture a sphere in a knot?

You and I decide to play a game: To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...
1 vote
1 answer
132 views

Least square assignment and hyperplanes

Let $S$ be a finite set of points in $\mathbb{R}^{d}$, $c(s) \in [0,1]$ such that $\sum_{s \in S} c(s) = 1$, $\rho$ continuous and non-vanishing probability distribution on $[0,1]^{d}$ and $\mu $ ...
0 votes
0 answers
52 views

Explicit calculation algorithm for distance function [duplicate]

I study differential geometry. Although there is a lot of study on the local theory, a global description lacks some explicit explanations. I mean, the study of surfaces describes curves, tangent ...
15 votes
1 answer
603 views

Acute triangles in "obtuse" polygons?

Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute? I conjecture ...
2 votes
1 answer
173 views

Four concyclic triangle centers

Can you prove the claim given below? Inspired by Lester's theorem I have formulated the following claim: Claim. Given any scalene triangle $\triangle ABC$ . Let $D$ be the reflection of incenter in ...
7 votes
2 answers
121 views

Completion of an Alexandrov space

Let $X$ be an incomplete Alexandrov space with sec $\ge -1$ in the sense that for any point in $X$ there exists a small neighborhood in which the four-points criterion is satisfied. Suppose $X$ is ...
3 votes
1 answer
111 views

Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim? Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
1 vote
0 answers
115 views

What is the maximum number of points on a plane such that some other point A is the nearest neighbor for all of them? [closed]

Point $A$ on the plane is given. How many other points can be placed on this plane under the condition that $A$ is the nearest neighbor for any of these points? I can think of a regular pentagon with $...
0 votes
1 answer
206 views

Does anyone know of an academic reference for this proof that the tangent to a hyperbola at a point bisects the angle between each focus to the point?

I am writing a report and as part of it I need to prove the property that for a point $P$ on a hyperbola, the tangent to the hyperbola at $P$ bisects the angle $\angle F_1PF_2$ where $F_1$ and $F_2$ ...
0 votes
0 answers
165 views

Adjunction formula for non compact surfaces

Let $M$ be a non compact complex surface and S an embedded compact Riemann surface in $M$. I already know how to show the following equality of fiber bundle: $$\Omega^2_{M}|S =\Omega^1_S \otimes N^*_S$...
1 vote
0 answers
171 views

Does there exist an isometry between a regular polygon and a circle?

In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...
4 votes
1 answer
286 views

Collinearity in bicentric polygons

Can you provide a proofs for the following two claims? Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear. Claim ...
5 votes
0 answers
149 views

Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?

It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure. Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ...
0 votes
0 answers
93 views

Distance Metric on a Polytope

Primary Question: Is it possible to define a distance metric on a polytope (or permutohedron in particular)? I am aware that neither is a smooth, Riemannian manifold; however, computer scientists have ...
5 votes
4 answers
1k views

On duality on finite projective planes

In nearly all (if not all) projective geometry texts I have bumped into the following theorem: "Principle of duality: If in a theorem in $\mathfrak{P}$ one switches the word point for the word ...
2 votes
1 answer
169 views

Metric projection on closed convex sets in Busemann space

I am looking for a proof of the following statement: Let $X$ be a complete Busemann space. For any point $x\in X$ and any nonempty closed convex set $A\subseteq X$, there is a unique $a\in A$ such ...
-2 votes
1 answer
128 views

A generalized norm function in $\mathbb{R}^n$ [closed]

We defined a new norm. The norm of $x \in \mathbb{R}^n$ is defined as $$ N_P(x) = \min \{t \geq 0 : x \in t\cdot P\} \enspace,$$ where $P$ is a centrally symmetric and convex body centered at the ...
15 votes
2 answers
998 views

Is every connected metrizable locally path connected space a length space?

Does every connected metrizable locally path connected topological space $X$ admit a compatible metric $d$ so that $(X,d)$ is a length space? (Edit to correct definition: Recall that a metric space $(...
2 votes
1 answer
203 views

Is there a theory of partially-defined metric spaces?

Is there a theory of metric spaces in which the distance between a given pair of points need not be defined? I'm aware that there is a theory of partial metric spaces, but these deal with a different ...
2 votes
0 answers
95 views

Kernels with finite dimensional feature spaces

Suppose $x,y \in \mathbb{R}^n$ for some given fixed n. Consider a kernel $K(x,y) = f(\langle x, y \rangle)$, I'd like to know which functions $f$ admit a finite dimensional feature map. In other words,...
3 votes
0 answers
892 views

Definition of quasi-geodesics

I was going through the books Géométrie et théorie des groupes by Michel Coornaert, Thomas Delzant, Athanase Papadopoulos and Metric Spaces of Non-Positive Curvature by Martin R. Bridson, André ...
7 votes
2 answers
428 views

More general form of inequality?

I have proved a simple Lemma that I need for a larger result, and I was wondering whether it is actually another more famous result in disguise. The lemma says that for any set of vectors in $\mathbb{...
7 votes
2 answers
1k views

Example of non-closed convex hull in a CAT(0) space

this is related to this question but is simpler, and hopefully is well-known. There are a number of references that say that the convex hull of a collection of points in a CAT(0) space need not be ...
2 votes
1 answer
142 views

Rolling wheel unicycle knots

Let $K$ be a knot, and $K(t)$ a parametrization of a space curve that realizes $K$. Roll a wheel $W$ of radius $r$ on $K(t)$ so that $W$ remains in the tangent-normal plane. Now track the wheel's ...
0 votes
1 answer
142 views

Cauchy's rigidity theorem

Newbie here, I'm studying the proof of Cauchy's rigidity theorem, but couldn't find any good resources. I read the chapter about it Proofs from THE BOOK, but it's really brief and I was not able to ...
2 votes
0 answers
197 views

Polyhedron - sphere intersection

newbie here. I'd like to ask you, if you know some brief, but somewhat solid proof of a convex polyhedron and a sphere centered at one of its vertices (with small enough radius, so it intersects only ...
1 vote
0 answers
48 views

Deployment and dispersion in triangular regions

Definitions (from C. Stanley Ogilvy's 'Tomorrow's Math'): Deployment: To place a specified number $n$ of points (stations) in a region such that the maximum distance of any point in the region from ...

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