Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,405 questions
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Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$
Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...
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2
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847
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What is known about polyhedra nets that allow overlapping?
It is an open problem that the net of any convex polyhedron can be unfolded onto a flat plane with no overlapping. Is anything known if we allow x faces to overlap? For example, is it known if any ...
- 2,615
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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\...
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A generic metric on $X\cup\mathbb Z$
$\newcommand\abs[1]{\lvert#1\rvert}$Let $(X,d_X)$ be a countable metric space such that $X\cap\mathbb Z=\{0\}$.
Problem. Is there a metric $d$ on the union $Y=X\cup\mathbb Z$ such that
$d(x,y)=d_X(x,...
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434
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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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Finding maximal prefix of a simple curve
Let $S$ be a simple curve. I want to determine maximal prefix of $S$ contained in a unit circle. Is this possible, or has it perhaps already been solved in the past, and I am just unable to find an ...
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Does complexified isometry group act transitively on tangent bundle of compact Riemannian manifold?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$Let $ g $ be the round metric on the sphere $ S^n $. Since $ S^...
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Rigidity for convex surfaces in elliptic/hyperbolic space
From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact ...
- 115
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A criterion for loxodromicity in Gromov-hyperbolic spaces
Recall that an isometry of a Gromov-hyperbolic space $X$ is called loxodromic if it has exactly two fixed points on the Gromov boundary $\partial X$, one being "attracting" and the other &...
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Can prolates overlap more easily than oblates?
Context:
When modeling anisotropic particles, the two common types of shapes of interest are cylindrical and disk-like particles. For simplicity let us say we model these as prolates and oblates ...
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Alexandrov's rigidity in higher dimensions
If $\Phi_1,\Phi_2$ are convex polyhedra in $\mathbb{R}^3$ such that the sets of outer normals to facets coincide, but $\Phi_1$ is not a translate of $\Phi_2$, then there exist two corresponding ...
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Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann's approach)
This question is a cross post from Math.SE. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I also have found this related question, ...
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Length of simple closed curve in half-translation surface
Let $R$ be a Riemann surface of genus $g\ge 2$ and $q$ an holomorphic quadratic differential on $R$. Together they determine a semi-translation structure: an atlas on $X$ such that its changes of ...
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Are metric isometries smooth at the boundary?
Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...
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607
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Upper bound for the number of integral points in a convex set
Let $K \subset \mathbb{R}^3$ be a bounded convex set such that the points with integer
coordinates in $K$ are not all coplanar. Is it true that $|K \cap \mathbb{Z}^3| \leq 6{\rm Vol}(K) + 3$?
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The Minkowski sum of two curves
Let $\gamma$ be a continuous curve in the complex plane without self-intersections and let $\lambda$ be a complex non-real number less than 1 in modulus. Put $\gamma'=\lambda\gamma$.
Question. Is it ...
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non-rigidity of interior points in polyhedral triangulations?
It's well-known that any compact polyhedron $P$ in $\mathbb{R}^n$ (we talk about piecewise-linear setting there, i.e. $P$ is a finite union of compact convex polytopes) can be triangulated into (...
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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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359
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Cone unfolding of space curves
There is a natural length-preserving operation which transforms any rectifiable space curve $\gamma\colon [a,b]\to R^n$ into a planar curve $\tilde\gamma \colon [a,b]\to R^2$. This operation, which ...
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Minimum area of the convex hull of the union of a parallelogram and a triangle
This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of ...
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What is the distance between two points on the Berger metric of the squashed three-sphere?
The Berger metric on a "squashed" three-sphere is given (in Euler coordinates) by
4 $ds^2 = \lambda^2 (d \tau + \cos \theta d \phi)^2 + d \theta^2 + \sin^2 \theta d \phi^2$.
See for example Eq. 1....
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Objects whose morphisms are Lipschitz maps
I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz").
The answer seems pretty clear, and proceeds like the definition of manifolds:
1) If $...
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153
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Above/below directed graph on cells of arrangement of lines
This question concerns the structure of a directed graph
built on the cells of an arrangement of lines.
My basic question is whether this graph has been
studied before, perhaps in another guise. I ...
- 151k
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550
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Approximating a real by a ratio of primes
Let $x$ and $y$ be positive reals in $(0,1)$ with $x < y$ and $y-x =\epsilon$.
I seek smallest primes $p$ and $q$ such that
$$x \le \frac{p}{q} \le (x+\epsilon) = y \;.$$
Q. What upper bound $u(...
- 151k
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260
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Convex hulls of quasi-convex sets in proper CAT(0) spaces
Let $A$ be a quasi-convex set in some proper CAT(0) space, $X$, and let $\mbox{Hull}(A)$ be the intersection of all convex sets containing A. Can we conclude that $\mbox{Hull}(A)$ is in some bounded ...
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648
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Maximal volume of a simplex inscribed in a spherical cap
Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\varepsilon)$ be the spherical cap with height $\varepsilon$ I am interested in the quantity
$$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in }...
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Escher, Conway, Kali, etc.
One can express the symmetry types of, say, Escher's "Circle Limit" prints using
Conway's orbifold notation, best known in the context of symmetries of Euclidean
plane patterns.
For example, Circle ...
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666
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What is the Cheeger constant of a cubical subset of the cubic lattice?
The Cheeger constant of a finite graph measures the "bottleneckedness" of the graph, and is defined as:
$$h(G) := \min\Bigg\lbrace\frac{|\partial A|}{|A|} \Bigg| A\subset V, 0<|A|\leq \frac{|V|}{2}...
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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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Iterating projections to random halfspaces
Consider the following process:
Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \...
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Rational stable translation length
Let $G$ be a finitely generated group and $S$ a finite generating set and consider the word metric associated to $S$.
If $g\in G$, define its stable translation length as $l(g)=\lim_n \frac{d(e,g^n)}{...
- 2,090
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438
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An isoperimetric type of inequality in terms of Wasserstein distance/Optimal transport
Let $A \subset \mathbb{R}^n$ be a region having the same volume as an $n$ dimensional ball $B^n_R$ with radius $R$ centring at the origin.
Isoperimetric inequality says:
$ Vol_{n-1} \partial A \geq ...
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Covering the plane with line segments with local hexagonal constraints
Can we characterize the following kinds of plane coverings? (Open-ended, but provide some description more "useful" than the constraints given.) For a more answerable question, is there an effective ...
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Comparison of angles in Alexandrov space
Let $X$ be a finite dimensional Alexandrov space with curvature bounded below. Let $p\in X$ be a fixed point.
Is it true that for any $\varepsilon >0$ there exists $\delta>0$ such that for any $...
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Regions on a sphere that avoid a fixed point set
Let $P$ be a finite set of points on a unit-radius sphere $S$
in $\mathbb{R}^3$.
Treat $P$ as a fixed pattern that can be rigidly slid
around $S$ as a unit (no reflection).
Let $R$ be a subset of $S$....
- 151k
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Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$
The equivalence I describe below is well-known, but I'd like a simple standard reference for it.
Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a ...
- 2,210
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Nonexpansive multi-valued maps in $\ell^2$
Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
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Sandwiching ellipses between planar convex bodies
Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that
$K' \subset L$
No ellipse $E$ satisfies $K' \subset E \subset L$
I am also ...
- 4,653
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Lonely globe trotters
In analogy with the lonely runners conjecture,
imagine "globe trotters" each traveling on a longitudinal great circle at different
(constant, positive) speeds. Each "trotter" ...
- 151k
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Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets
We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...
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A spherical geometry claim related to the perspective 3-point problem
I have a simple claim in spherical geometry that has come out of my research into the so-called "perspective 3-point (pose) problem."
Here it is:
Fix three (distinct) great circles on the ...
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Tiling space with supertile of hypercube unfoldings
Two students in my class
asked and answered what might be a novel question.
It is well known that the cube has exactly $11$ edge-unfoldings
(or "nets"), as shown below:
(Image from ...
- 151k
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A weak analogue of smooth manifolds (reformulated)
It is widely known that $C^1$ manifolds are topological spaces locally homeomorphic to Euclidean spaces and possessing $C^1$ chart-converters. They have a tangent space at every point, regarding as ...
- 1,543
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495
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A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel
I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:
Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...
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Set of unit vectors such that among any three there is an orthogonal pair
I was fascinated by the solutions of Problem 8 of the IMC 2021 contest, which can be summarized as:
Theorem 1. Let $v_1,\dotsc,v_N$ be distinct unit vectors in $\mathbb{R}^n$ such that among any three ...
- 105k
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Extending continuous injective curves both continuously and injectively
Let $X$ be a topological space.
Let $\gamma:[a,b]\to X$ be continuous and injective.
$\gamma$ is said to be "openly extendable" if there is $[a,b]\subset (a',b')$ and a continuous and ...
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346
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The space of $p$-adic norms
The 1963 paper by Goldman and Iwahori The space of $p$-adic norms deals with the space of norms on a finite dimensional vector space $E$ over a locally compact complete discrete valuation field $K$. I ...
- 433
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494
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Applications of the co-area formula
Kirchheim [2] generalized the classical area formula to the case of Lipschitz mappings into metric spaces. Ths paper is well known and widely cited. The area formula is a special case of the co-area ...
- 28k
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Hanging a cube with string
This is a variation on a (much) earlier MO question, Hanging a ball with string.
Here instead the task is to arrange a net of string to hang
a unit cube. Assume:
The string is inelastic.
There is no ...
- 151k