Questions tagged [mg.metric-geometry]

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

Filter by
Sorted by
Tagged with
4 votes
0 answers
64 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) ...
Jochen Wengenroth's user avatar
-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?
asa's user avatar
  • 3
-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?
asa's user avatar
  • 3
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 ...
Joel David Hamkins's user avatar
9 votes
1 answer
299 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, ...
Taras Banakh's user avatar
  • 40.7k
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 ...
Nandakumar R's user avatar
  • 5,401
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 ...
Danny's user avatar
  • 1
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, \...
Marty's user avatar
  • 13.1k
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 ...
M. Winter's user avatar
  • 12.5k
-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 ...
MMH's user avatar
  • 129
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&...
Kacper Kurowski's user avatar
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\...
Christian's user avatar
  • 779
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)$...
Vladimir Zolotov's user avatar
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 ...
M. Winter's user avatar
  • 12.5k
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(...
Gesh's user avatar
  • 105
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....
Stefan Vujic's user avatar
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 $\varepsilon$-...
asv's user avatar
  • 21.1k
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 ...
GRquanti's user avatar
  • 111
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 ...
zemora's user avatar
  • 545
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 ...
RandomTensor's user avatar
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 $\mathbb Z^n$?
WKC's user avatar
  • 646
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 $x\in \mathbb R^d$, ...
GJC20's user avatar
  • 1,210
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 $n$ ...
Nandakumar R's user avatar
  • 5,401
-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 ...
mathoverflowUser's user avatar
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+\...
asv's user avatar
  • 21.1k
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 ...
asv's user avatar
  • 21.1k
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-...
David Bryant's user avatar
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 ...
Nandakumar R's user avatar
  • 5,401
3 votes
1 answer
75 views

Embedding a countably infinite metric space in $\ell^2(\mathbb Z_+)$

Suppose $(X,d)$ is a countably infinite set endowed with a metric $d$ that satisfies the following condition: Every finite subset of $X$ with the induced metric is isometric to a subset of some ...
Daniel Asimov's user avatar
2 votes
0 answers
74 views

Nested convex hulls in Hadamard manifold

Let $F$ be a finite set in a Hadamard manifold $H$, and $W\supset F$ is its neighborhood. Is it true that the closure of the convex hull of $F$ lies in the interior of the convex hull of $W$? ...
Anton Petrunin's user avatar
0 votes
0 answers
38 views

What is the dual of a hyperbolic configuration of points?

Let $C_n$ denote the configuration space of $n$ distinct points in hyperbolic $3$-space $\mathcal{H}$. If $\mathbf{x} := (\mathbf{x}_1, \dots, \mathbf{x}_n) \in C_n$, where $\mathbf{x}_i \in \mathcal{...
Malkoun's user avatar
  • 4,981
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....
Nandakumar R's user avatar
  • 5,401
3 votes
0 answers
88 views

Sampling uniformly from the convex cone

Let $n$ vectors of dimension $d$ (e.g., $n = 100$, $d = 10000$), each with infinity norm of $1$, be given. The conic combination of those $n$ vectors generates a convex cone. How to uniformly sample ...
mathhamcs's user avatar
3 votes
0 answers
148 views

What are the measure of the volume and boundary (and other quermaß measures) of the positive semidefinite matrices?

Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q)...
Gro-Tsen's user avatar
  • 29.8k
1 vote
2 answers
190 views

A triangle comparison in CAT(0) spaces

Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in ...
Mohammad Ghomi's user avatar
0 votes
0 answers
46 views

Which planar convex region with specified area and perimeter maximizes/minimizes Moment of Inertia?

By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C. Question: For specified ...
Nandakumar R's user avatar
  • 5,401
2 votes
1 answer
163 views

Forming paper bags that can 'trap' 3D regions of max surface area

An existence question based on 'Trapping' 3D regions with sheets of paper. Given a sheet of paper S that is a planar convex region, one tries to form a 'closed bag' that contains a connected ...
Nandakumar R's user avatar
  • 5,401
4 votes
1 answer
128 views

Characterization of convexity by connectedness of hyperplane sections

Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$. Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...
Dmitrii Korshunov's user avatar
2 votes
0 answers
44 views

Instances of c-concavity outside of optimal transport?

Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
Brendan Mallery's user avatar
2 votes
1 answer
150 views

Osculating sphere at point of maximal curvature lies to one side

I'm looking for the higher dimensional version of this post, which says that given a curve $\gamma \subseteq \mathbb{R}^2$, the osculating circle will lie to one side of the curve at points of maximal ...
JMK's user avatar
  • 301
0 votes
0 answers
68 views

On 'Width Equalizers' of planar convex regions

Definitions: The least width of a 2D convex region C is the least distance between any pair of parallel lines that both touch the boundary of C (in what follows, we refer to this quantity as simply '...
Nandakumar R's user avatar
  • 5,401
3 votes
1 answer
147 views

Equivalent definition for Skorokhod metric

I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$: $$ d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
user1598's user avatar
  • 177
7 votes
2 answers
281 views

Étendue measure of the set of lines between two Euclidean balls

Let $d>0$ and $r_1,r_2>0$ such that $r_1+r_2 < d$. Consider two (say, closed) balls $B_1,B_2$ in $\mathbb{R}^m$ having radii $r_1,r_2$ and whose centers are at distance $d$. Let $C$ be the ...
Gro-Tsen's user avatar
  • 29.8k
1 vote
0 answers
79 views

$L^p$-compression of metabelian groups

Question: Is there a metabelian group, so that for some $\epsilon >0$ and all $p \in [1, \infty[$ the [equivariant] compression exponent in [any] $L^p$-space is bounded by $1-\epsilon$ (bound does ...
ARG's user avatar
  • 4,342
3 votes
0 answers
110 views

Reference request: Carathéodory-type theorem for convex hulls of closed sets

I'm looking for a reference for the following theorem. Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
Tom Leinster's user avatar
  • 27.2k
1 vote
0 answers
78 views

Can the volume of a neighborhood of the cut locus be arbitrarily small?

Let $(M^n,g)$ be a complete, $n$-dimensional Riemanian manifold without boundary, maybe non-compact. Let $p\in M$ be a point, and $C_p$ the cut locus. It's known that $C_p$ has Hausdorff dimension $\...
mathmetricgeometry's user avatar
5 votes
0 answers
74 views

Complexity and length

Suppose we define continuous piecewise linear functions $f$ on $[0,1]$ using your favorite programming language, or by finite automata, or by any other suitable machine. Define the complexity $H(f)$ ...
Dmitrii Korshunov's user avatar
0 votes
0 answers
58 views

Comparing partitions of a given planar convex region into pieces with equal perimeter and pieces of equal width

We continue from Cutting convex regions into equal diameter and equal least width pieces. There we had asked for algorithms to partition a planar convex polygon into (1) $n$ convex pieces of equal ...
Nandakumar R's user avatar
  • 5,401
3 votes
0 answers
87 views

Minkowski problem for polytopes: the origin of necessary condition

Minkowski's uniqueness theorem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets. Theorem (Minkowski). Let $A_i$ be positive faces areas ...
Alexey Ustinov's user avatar
0 votes
0 answers
89 views

Bounding the area of the image of a set by product of maximum of lengths

Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$. My question feels ...
JustSomeGuy's user avatar

1
2 3 4 5
79