Questions tagged [mg.metric-geometry]

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

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4 votes
0 answers
56 views

Quotients in categories of metric spaces

There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (=non-expansive or contractive) ...
-4 votes
0 answers
42 views

Non-isomorphic geometric objects obtained by cutting the Möbius band [closed]

How many non-isomorphic geometric objects can be obtained by cutting a Möbius band while keeping parallel on the substrate?
4 votes
1 answer
138 views

Reference request: Fréchet embedding

Given a separable metric space $(X,d)$, we have an isometric embedding $\iota:X\to\ell^\infty$ given by taking $(x_n) _{n \ge 0}$ to be the countable dense subset and sending $\iota(x)_n=(d(x,x_n) - d(...
6 votes
1 answer
546 views

When is the cut locus a finite tree?

Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for ...
9 votes
1 answer
298 views

Is the group of translations of an affine plane always commutative?

$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, ...
15 votes
0 answers
737 views

Are all these groups CAT(0) groups?

Given a geodesic metric space $X$ together with a choice of midpoints $m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$). Assume furthermore, that the following nonpositive curvature ...
-1 votes
0 answers
54 views

Cube containing nine unit cubic lattices [closed]

What is the volume of the smallest cube containing nine unit cubes?
18 votes
1 answer
1k views

Does greedy circle packing exhaust the measure of every bounded open set in the plane?

The greedy circle packing of a bounded region in the plane is the result of placing at each stage the largest possible disk into the region that remains uncovered. The greedy circle packing of a ...
10 votes
1 answer
292 views

Is a convex polyhedron determined by its edge lengths and angular defects?

Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$. The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$. Question: Is a ...
7 votes
1 answer
324 views

A corollary of the affine Desargues axiom

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
1 vote
0 answers
37 views

Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces

Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
5 votes
1 answer
384 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 ...
15 votes
4 answers
1k views

Geodesics in $\mathbb{R}^2 \times \mathbb{S}^1$ under "segment" metric

Represent the position of a unit-length, oriented segment $s$ in the plane by the location $a$ of its basepoint and an orientation $\theta$: $s = (a,\theta)$. So $s$ can be viewed as a point in $\...
23 votes
3 answers
2k views

Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$. Say that $S$ is an integer-distance set if every pair of points in $S$ is separated by an integer Euclidean distance. ...
0 votes
0 answers
31 views

Calculate the intersection volume of two spherical caps on the same sphere [migrated]

My question is the same as this question except that I am looking for the intersection volume of two caps instead of the area. Given a sphere with radius $R$ that has two spherical caps with base ...
4 votes
0 answers
103 views

Curiosity about "conditional trig identities"

Perhaps this should be cross-posted on Math Stackexchange, but it came up in the context of some research mathematics (quaternion orders, etc.) In this context, I have three angles $\alpha, \beta, \...
10 votes
1 answer
252 views

Do triple-linked graphs exist?

Lets say that a finite simple graph $G$ is (intrinsically) fully triple-linked if for each embedding of $G$ into $\Bbb R^3$ we can find three disjoint cycles $C_1,C_2,C_3\subset G$ whose embeddings ...
16 votes
0 answers
288 views

Realization spaces of 3-dimensional polytopes with fixed face areas

It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible. A proof of this theorem can be found for instance in ...
-2 votes
1 answer
60 views

Inner Products of Elements in Spherical Cap [closed]

I am interested in understanding what is the lowerbound on the inner products of two elements of a sphere. Based on my intuition in dimension 2, I come up with the following conjecture. I appreciate ...
9 votes
0 answers
208 views

Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces

Is it know which metric spaces $M$ do have the following property: there is $\varepsilon>0$ and a maximal $1$-separated set which is $(1-\varepsilon)$-dense? In other words, when does at set $S\...
1 vote
0 answers
90 views

A claim on the largest area circular segment that can be drawn inside a planar convex region

This post adds a little to To find the longest circular arc that can lie inside a given convex polygon A circular segment is formed by a chord of a circle and the line segment connecting its endpoints....
2 votes
0 answers
52 views

For riemannian manifolds, how close can a mapping from atlas be to an isometry?

Let $(M, g)$ be an $n$-dimensional $C^k$ (or $C^\infty$) Riemannian manifold. On $M$ we can define metric $d_g$ as the infimum of lengths of curves that connect given two points. Fix $x \in M$ and $r&...
28 votes
3 answers
2k views

Is the ratio Perimeter/Area for a finite union of unit squares at most 4?

Update: As I have just learned, this is called Keleti's perimeter area conjecture. Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
1 vote
1 answer
86 views

To place copies of a planar convex region such that number of 'contacts' among them is maximized

A contact between two planar convex regions obviously happens either along a line segment or at a single point. Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...
3 votes
5 answers
1k views

Approximate solutions for trisecting the angle or squaring the circle

Hello all, it is well-known by transcendence results or Galois theory that famous geometric problems such as trisecting an angle or "squaring the circle" (i.e. given a disk of radius 1 construct a ...
2 votes
0 answers
66 views

Are 1-Wasserstein and 2-Wasserstein distances between multivariate normal distributions equivalent?

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by $$W^p_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
1 vote
1 answer
305 views

Geometry in $\mathbb{R}^n$: angle between projections of a rectangle

Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$. Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$. For ...
3 votes
2 answers
2k views

Cone angles for Riemannian metrics in polar coordinates

This is the simplest case of a question that's been bugging me for a while: say we have a Riemannian metric in polar coordinates on a $(2-d)$ surface: $$ g=dr^2+f^2(r, \theta;)d\theta^2, $$ such that ...
2 votes
1 answer
108 views

Binary codes with upper and lower bound on pairwise distance

The Gilbert-Varshamov bound provides a lower bound for codes of length $n$ with minimum pairwise distance (say $\frac{n}8$). If we wish for the codes to also have pairwise distances bounded above (say ...
0 votes
0 answers
51 views

Three annulus intersection problem [migrated]

Recently I faced a problem about intersection of three annuli. Imagine having three annuli same dimensions and you put them next to each other into triangular shape like putting together three circles....
27 votes
8 answers
5k views

Convex hull in CAT(0)

Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset. Is it true that convex hull of $K$ is compact? Comments: Convex hull of $K$ = intersection of all closed convex sets ...
1 vote
1 answer
115 views

Connectedness of fibers of almost Riemannian submersions

EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary. Given $f\colon M\to N$ be an $\varepsilon$-...
47 votes
4 answers
11k views

Volumes of n-balls: what is so special about n=5?

I am reposting this question from math.stackexchange where it has not yet generated an answer I had been looking for. The volume of an $n$-dimensional ball of radius $R$ is given by the classical ...
6 votes
1 answer
319 views

Stability of midpoints in CAT(0) spaces

Given a CAT(0) space $X$ and a compact, convex subset $A$ of $X$. One can define its midpoint $m(A)$ as the point, at which the following function attains its minimum. $f:A\rightarrow \mathbb{R}\qquad ...
10 votes
2 answers
666 views

Geometric construction of the fourth intersection points of two conics

In general, two conics in the plane intersect at most 4 points. Suppose three of those points are given as $A,B,C$. Then let $c_1$ be the conic passing through those three points and $D_1,E_1$. Let $...
1 vote
0 answers
62 views

Algorithm to generate configurations with kissing number 12

That the kissing number of a sphere in dimension 3 is 12 is well known. However, it is also known that there is a lot of empty space between the 12 spheres. I deduce (am I wrong?) that there are many ...
6 votes
0 answers
153 views

Does there exist a plane curve such that it has the heart curve as catacaustic?

Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$. The catacaustic ...
5 votes
0 answers
128 views

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
1 vote
2 answers
262 views

Does $\mathbb Z^n$ contain $A_n$?

Are there any positive integer $n > 3$ such that the root lattice $A_n$ is contained in $\mathbb Z^n$?
3 votes
1 answer
176 views

Dimension of Alexandrov space which is homeomorphic to a manifold

Let $M^n$ be a smooth manifold of dimension $n$. Let $M$ given a metric with curvature bounded below in the sense of Alexandrov which induces the original topology of $M$. It is true that the ...
0 votes
0 answers
51 views

Seek a partition of $\mathbb R^d$

Let $c>0$ be given. I look for $n\ge 1$ and a collection of closed subsets $(F_i: 1\le i\le n)$ such that $$\bigcup_{1\le i\le n} {\rm int}(F_i)= \mathbb R^d,$$ and for every $x\in \mathbb R^d$, ...
4 votes
1 answer
122 views

Does the Alexandrov angle define convex functions along geodesics in CAT(0) spaces?

Let $X$ be a CAT(0) space and suppose $a,b,c,d\in X$ satisfy $$ \max\{\angle_a(b,c),\angle_a(b,d)\}<\frac{\pi}{2}. $$ Let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)...
-2 votes
1 answer
122 views

Interpretation and validity of modified Heisenberg uncertainty principle in a metric context? [closed]

Considering the Heisenberg uncertainty principle, which states $\Delta x \cdot \Delta p \geq h$, I've explored a modified version by computing $(\Delta x + 1)(\Delta p + 1) \geq \Delta x \cdot \Delta ...
3 votes
1 answer
193 views

"Almost geodesics" in Riemannian manifolds which cannot be loops

Let $M,N$ be smooth Riemannian manifolds of the same dimension. Let $0<\varepsilon<\frac{inj(N)}{100}$. Let $f\colon M\to N$ be a smooth map such that for any $x\in M$ and any $v\in T_xM$ one ...
6 votes
0 answers
176 views

When is a distance space dominated by a metric space?

A distance space is a pair $(X,d)$ where $X$ is a set and $d:X \times X \rightarrow \mathbb{R}$ is a symmetric, non-negative map such that $d(x,x)=0$ for all $x \in X$. These are sometimes called semi-...
1 vote
1 answer
83 views

When "$(\varepsilon,\delta)$-geodesic" cannot be a loop?

EDIT: Let $M$ be a smooth compact Riemannian manifold. Let $\varepsilon,\delta>0$. I call a smooth curve $\gamma\colon [a,b]\to M$ an $(\varepsilon,\delta)$-geodesic if for any $t_1<t_2<t_1+\...
1 vote
1 answer
56 views

To maximize the volume of the polyhedron resulting from perimeter-halvings of a convex polygonal region

We add one more bit to Forming paper bags that can 'trap' 3D regions of max surface area (note: some possibly open related questions are also in the comments following the answer to above ...
2 votes
0 answers
115 views

Folding polygons into 'vessels'

This question is based on http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf Define an vessel as a convex polyhedron with one face removed - in other words, a vessel can be converted into a ...
3 votes
1 answer
879 views

Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$): $\min_{x\in U}\|x-y\|$. I believe the minimizer $x_{U}^{*}$ is ...
2 votes
0 answers
115 views

Integral geometric meaning of diameter

Let $X\subset \mathbb CP^n, n>2$ be a complex smooth algebraic hypersurface. Any hyperplane section $H\cap X$ is connected and has diameter $Diam(H\cap X)$ in the inner metric induced from the ...

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