# Questions tagged [mg.metric-geometry]

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

3,710
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### Symmetric Busemann G-spaces are homeomorphic to Euclidean spaces

For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$.
Definition: A metric space $(X,d)$ is called
$\bullet$...

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### Concurrencies determined by intersections of angle trisectors (and isogonal lines) in a triangle

The famous Morley’s theorem, states that in a triangle the interior angle trisectors, proximal to sides respectively, meet at the vertices of an equilateral. However the six trisectors meet at 12 ...

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### Sweeping out the disk: what comes out?

In 2008, Larry Guth gave a new proof of a theorem of Gromov about the min-max widths of the unit $n$-ball. This states that the $p$-parameter width $\omega_p(k,n)$ (of sweepouts with $k$-dimensional ...

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### Mcq question on product of length of chords [closed]

A unit circle has $n$ points on it such that the arcs made by these points are equal in size. Now draw edges b/w every pair of these n points. What is the product of length of these chords?
a) $1$
b) $...

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### Geometry of inner products between the unit vector and several given vectors

Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e.,
$$
\mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...

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### Are aperiodic monotiles generalizable to higher dimensions?

This question is motivated by a recently released paper written by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. It constructs the first topological disk that tiles the ...

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### Plateau problem in the disk: a question about geodesic nets

Consider given a finite collection of points along the boundary of the unit disk $D \subset \mathbf{R}^2$:
\begin{equation}
p_1,\dots,p_{2n} \in \partial D.
\end{equation}
We assume that these are all ...

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### Fitting a convex polytope with 𝑛 facets between two nested spheres

This is related to a research problem that is interested in approximation of spheres by convex polytopes.
Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...

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### A metric characterization of Hilbert spaces

In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...

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### Is Morley’s observation complete?

Morley’s observation states that in a triangle the intersections of trisectors proximal to a (triangle) side lie six by six on three triples of parallel lines that make angles of 60° with each other. ...

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### Inside-out dissections of solids

We add to Inside-out dissections of polygons - a generalization. The inside-out (fully inside-out) dissections are defined on pages linked there.
How does one inside-out dissect a tetrahedron into ...

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### About the number of faces of the conification of a polytope

Let $P\subset\mathbb{R}^n$ be a polytope of dimension $(n-1)$ such that the origin $\vec{0}\not\in\text{Aff}(P)$, where $\text{Aff}(P)$ denotes the affine hull of $P$ in $\mathbb{R}^n$. Now, we ...

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### Vector measures as metric currents

Currents in metric spaces were introduced by Ambrosio and Kirchheim in 2000 as a generalization of currents in euclidean spaces. Very roughly, a principle idea is to replace smooth test functions (and ...

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### A metric characterization of the real line

Is the following metric characterization of the real line true (and known)?
A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real ...

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### non-negative curvature condition for polyhedral manifolds

A polyhedral manifold P, i.e, a topological manifold with a triangulation where each simplex is isometric to a simplex in Euclidean space (other constant curvature spaces are allowed), is said to have ...

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### Smallest trapeziums containing a given convex n-gon

Question: Given a planar convex $n$-gon $C$, to find the smallest area / smallest perimeter trapezium (trapezoid) - a convex quadrilateral with at least one pair of mutually parallel edges - that ...

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### If $X,X'$ have the same $\varepsilon$-packing numbers and $f:X \to X'$ surjective $1$-Lipschitz, then $f$ is an isometry

Let $(X, d)$ be a compact metric space.
We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$-covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, ...

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### Are two metric spaces isometric if they have the same $\varepsilon$-covering and $\varepsilon$-packing numbers for all $\varepsilon>0$?

Let $(X, d)$ be a compact metric space.
We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$-covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, ...

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### Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$?

Let $(E, d)$ be a metric space. For $\varepsilon>0$, we define two notions of $\varepsilon$-covering number as follows, i.e.,
$N_\varepsilon^o (E)$ is the smallest number of open balls whose radii ...

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### Cutting triangles into triangles with equal longest side

This post elaborates on a specific instance of Cutting convex polygons into triangles of same diameter .
Question: For any integer n, can any triangle be cut into n non-degenerate triangles all of ...

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### Triangle drawn in region bounded by $x$-axis and polynomial with all real roots: supremum of ratio of areas?

A triangle is drawn in a region bounded by the $x$-axis and a polynomial curve with all real roots, with one side of the triangle on the $x$-axis, as shown in this example:
What is the supremum of ...

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### $1$-Lipschitz map from hyperbolic to Euclidean plane

I'm trying to find a reference to the following statement.
Define a function $f$ from the hyperbolic plane (in the Poincaré unit disc model using polar coordinates) to the Euclidean plane (using polar ...

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### Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isometry?

I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an ...

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### Is there a bicyclic irregular pentagon in integers?

Is there a bicyclic irregular pentagon in integers, i.e. is there a pentagon, the length of each side is integer and unique such that it has a circumcircle and an inner circle as well?
If it does ...

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### Efficiently determining surface intersections along a line segment

Background
In general, I know how to determine the points of intersection between a surface and a line. In my case, I may have a large number of defined surfaces that may (or may not) intersect each ...

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### What is the example of a circle being filled with congruent tiles (not pie slices), with no overlap of the tiles and and no space left?

I think I read somewhere that at one time it was thought the only way to lay tiles that would fill a circle with no overlap of the tiles and no exposed space in the cirlce, was to lay pieces that ...

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### A problem about matrix

Assume that $\boldsymbol{v}_{\boldsymbol{1}}, \ldots, \boldsymbol{v}_{\boldsymbol{n}} \in \mathbb{R}^n$ satisfy $\forall i, j \in[n],i \neq j,\left\langle\boldsymbol{v}_{\boldsymbol{i}}, \boldsymbol{v}...

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### Does the edge-graph of a centrally symmetric polytope determine which vertices are antipodal?

Given two origin symmetric convex polytopes $P_1$ and $P_2$ (that is $P_i=-P_i$) with the same edge-graph, but potentially of different dimensions and combinatorial types.
Let $\phi: G_{P_1}\to G_{P_2}...

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### Least number of squares of size N that a set of R rectangles can occupy

Given a set $R$ of rectangles of different positive integer sizes, and any number of squares of the same size $N\in\mathbb{N}$, what's the least number of squares $C$ that all the rectangles together ...

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### Visualisation of general 3x3 matrices, with applications to the pedagogy of linear algebra?

I've got a method for visualising non-zero $2 \times 2$ real matrices (modulo non-zero scalar factor) using the fact that:
Nonnegative determinant matrices (modulo non-zero scalar factor) are in 1-to-...

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### Embedding a graph into Euclidean space

I want to find a map $v\mapsto \tilde v$ from the vertex set of a connected infinite graph $\Gamma$ to a Euclidean space that meets the following two conditions:
there is $\varepsilon>0$ such that ...

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### Conformal changes of metric and normal coordinates

Suppose that $(M,g)$ is a smooth Riemannian manifold of dimension $n\geq 2$. Let $p\in M^{\textrm{int}}$. Does there exist a small $\delta>0$ and a smooth function $c>0$ such that for the ...

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### The closest ellipse and circle to a given triangle - 2

We add a little more to The closest ellipse to a given triangle.
The above linked discussion used the Hausdorff distance to quantify how close two planar convex regions are.
In an earlier post - ...

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### Isolated maxima for sum of distances of points on a manifold

Let $X$ be a closed Riemannian manifold and consider the function $f_n : X \times \cdots \times X \to \mathbb{R}$ where the domain of $f_n$ is the $n$-fold cartesian product of $X$ and where $f_n(p_1,....

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### Does the surface area of the unit Lp ball go to zero for all $p < \infty$?

We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...

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### Does this "join-like complex" of $K_5$ and $K_3$ embed in $\Bbb R^4$?

Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$.
Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ ...

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### Is a facet always a maximal area section of a simplex?

Let $T\subset \mathbb{R}^n$ be a fixed simplex, $H\subset \mathbb{R}^n$ be a variable affine hyperplane. Is it true that the maximal area (i.e. the $(n-1)$-dimensional volume) of $T\cap H$ is attained ...

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### Can a polytopal graph be "centrally symmetric" in more than one way?

Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$.
The central symmetry of $P$ induces an involutory ...

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### On tiling the plane with non-congruent, equal area triangles with each edge having a unique length

Ref: Tiling with incommensurate triangles shows an approach for tiling with incommensurate triangles - all sides and all angles unique and also with different areas - with the perimeters of the tiles ...

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### Trigonometry/spherical angles/minimum-least-squares [closed]

An issue from 3D tessellated geometry: Find the direction vector of a plane that minimizes the silhouette of a set of triangles. To say it another way, find the direction vector that is most ...

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### Do the heights of an acute triangle intersect at a single point (in neutral geometry)?

A well-known result of the Euclidean planimetry says that the heights of any triangle have a common point called the orthocentre of the triangle. This result is not true in neutral geometry (i.e., ...

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### Kissing behavior of planar regions

This post reworks a question that was stated in a slightly different form at Convex region $C$ with least kissing number of copies of $C$.
Background: Given a 2D region $C$ (not necessarily convex), ...

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### Finding angle with geometric approach [closed]

I would like to solve the problem in this picture:
with just an elementary geometric approach. I already solved with trigonometry, e.g. using the Bretschneider formula, finding that the angle $ x = ...

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### Uniformly continuous homotopy equivalence

Suppose $M$ and $N$ are complete metric spaces and $f, g: M \to N$ are uniformly continuous maps between them with common modulus of continuity $m$. Further suppose $f$ and $g$ are homotopy equivalent....

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### The closest ellipse to a given triangle

Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given a general triangle T, to ...

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### Is the hypotenuse operation associative in every Tarski plane?

By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...

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### Is an equilateral triangle constructible in a Tarski plane?

By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...

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### Concentration of volume towards the boundary

Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let
$$G_\varepsilon:=\{x\in G\mid B_\varepsilon(x)\subseteq G\}$$
be the set of all ...

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1
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45
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### On 'axiality' of planar convex regions

Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry)
Consider an alternative definition of axiality as follows: For a convex region C, consider a ...

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### Does every Tarski plane embed into a 3-dimensional Tarski space?

By a Tarski space I understand a mathematical structure $(X,B,\equiv)$ consisting of set $X$, a betweenness relation $B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ ...