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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Does every positive-definite integral lattice admit an angle-preserving homomorphism into $\Bbb Z^n$ for some $n$?
Some initial clarifications
By lattice I mean an additive subgroup of $\mathbb R^n$ which is isomorphic to $\mathbb Z^n$ and has full rank (i.e. spans $\Bbb R^n$ when considered as set of vectors). A ...
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Gromov-Hausdorff convergence for locally finite metric spaces
This question might be very easy, but I am little confused by the Gromov-Hausdorff convergence.
My situation is the following: I have a fixed set $X$ which is finite or countable; on it I have ...
- 6,034
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Morphism between projective varieties
Let $f:X \rightarrow Y$ be a morphism between two smooth projective varieties $X,Y$ which are defined over an algebraically closed field $k$. I am looking for some criteria which guaranties the ...
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Rep-tiles of order 2
A 2-rep-tile is a geometric shape that can be partitioned into exactly 2 smaller (dilated) copies of itself. Although there are many rep-tiles of higher orders, the only 2-rep-tiles I could find are:...
- 3,549
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Path length of ball on tilted, perforated plane
Imagine that an $\epsilon$-radius hole is punched in the plane centered
on every integer-coordinate point.
Now a point "ball" is dropped on the plane at a random spot $p$.
If $p$ has not already ...
- 151k
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Tessellating $\mathbb{R}^n$ by bricks.
Let us call the $\ell_1$-product of intervals $[0,k_1]\times...\times [0,k_n]$ a brick of size $k_1+...+k_n$. Consider a tessellation $T$ of $\mathbb{R^n}$ by (shifted) bricks so that every point ...
user6976
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What is Kept Fixed for Flexible Spheres
For background to this question much recent exciting related things, see this videotaped lecture by Alexander Gaifullin.
Consider a triangulation $K$ of a two-dimensional sphere and consider maps ...
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Do maximal polyhedra have algebraic volume?
Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number?
Update: What can ...
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Circle-arc number of a knot
I would like to build knots in $\mathbb{R}^3$ from arcs
of unit-radius (planar) circles, joined together at points where
the tangents match. Thus the knot will have
curvature $1$ at all but the ...
- 151k
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Is every knot unavoidable in the embeddings of some graph?
Is it the case that, for any given knot $K$,
there exists some graph $G$ whose every embedding into $\mathbb{R}^3$
(or into $\mathbb{S}^3$)
contains a cycle that realizes $K$?
I know the famous ...
- 151k
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Polyhedron not circumscribed about a sphere
Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
My teacher ...
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Relating curvature and torsion of a connection to those of a curve
I'm currently trying to relate two descriptions of the curvature and torsion of a connection and am running into some confusion.
I know that an affine connection $A$ on an $n$-dimensional manifold $M$...
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Geodesic path on the unit sphere with the sup norm
Let $X$ be the unit sphere of $\mathbb{R}^n$ with the sup norm, i.e. $X=\{x\in\mathbb{R}^n: \|x\|_{\infty}=1\}$. Let the metric $d$ on $X$ be the geodesic metric induced by the sup norm, i.e for any ...
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Geometric interpretation of matrix minors
i was recently interested in geometric interpretation of various notions showing up in linear algebra because in most cases linear algebra with geometry courses focus too much on linear algebra not ...
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Mean maximum distance for N random points on a unit square
Following up on Mean minimum distance for N random points on a one-dimensional line and Mean minimum distance for N random points on a unit square (plane), I have the following questions.
Given N ...
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Largest regular $k$-simplex inscribed in a $d$-cube, $k < d$
The largest (by edge length) regular simplex inscribed in a unit cube
is well known in $\mathbb{R}^2$ and $\mathbb{R}^3$:
Image sources:
left: NMSU,
right: Mathworld.
A recent Amer ...
- 151k
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Monotonicity of Loewner ellipsoid?
Given two $0$-symmetric convex bodies $K \subset L \subset \mathbb{R}^n$, is it true that the Loewner ellipsoid of $K$ is contained in the Loewner ellipsoid of $L$?
I have just finished proving a ...
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Does a compact contractible metric space have a point that is fixed by all isometries?
Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries.
Question: Is there a point $x\in X$ fixed by all $\phi\in\...
- 13.6k
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Optimal $\delta$ for Gromov's $\delta$-hyperbolicity of the hyperbolic plane
What is the minimal $\delta$ such that the hyperbolic plane is $\delta$-hyperbolic, in the sense of the four point definition of Gromov?
Four point definition of Gromov: A metric space $(X, d)$ is $\...
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Origin of term Ahlfors-David regular
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...
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How many positions of a tiling polygon can occur simultaneousy?
Let $T$ be a polygon which tiles the plane. For an instance of $T$ (mirrored or not), call the set of its translates a position of $T$.
My question:
How many different positions can occur in ...
- 13.4k
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$G$-structures of finite type.
A $G$-structure $\pi : B_G \rightarrow M$ is said to be of $finite$ $type$ if $\mathfrak{g}^{(k)} = 0$ for some $k \in \mathbb{N}$, where $\mathfrak{g}^{(k)}$ denotes the $k$th prolongation of the Lie ...
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Is there a compass and straightedge construction of parallel lines in hyperbolic geometry?
Is there a compass and straightedge construction of parallel lines in hyperbolic geometry?
That is, given a line and a point not on the line, construct a line parallel to the given line.
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Translation lengths in CAT(0) spaces
Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently ...
- 8,401
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Monochromatic point sets in two-colored plane
Which are the configrations $P\subset \mathbb{R}^2$ of points, such that the following property holds:
Property M (for Monochromatic): Every two-coloring of $\mathbb{R}^2$ contains a monochromatic ...
- 10.7k
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A characterization of metric spaces admitting a bi-Lipschitz embedding into a Hilbert space?
Theorem (??) derived in this MO-post from Schoenberg's theorem yeilds a "bipartite" characterization of metric spaces that admit an isometric embedding into a Hilbert space. This Theorem (??)...
- 42k
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Circles avoiding rational points of height $\le h$
Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$)
of radius $r < 1$ avoid all rational points
of height $\le h$?
A rational point is a point all of whose coordinates are ...
- 151k
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Is this a metric on the Grassmannian Manifold?
Let $m>n$ and consider the Set
$$S_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I_n \}.$$
Does the function $d\colon S_{m,n} \times S_{m,n} \rightarrow \mathbb{R}$ defined by
$$d(A,B)=\sqrt{1-\...
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Get Largest Inscribed Rectangle of a Concave Polygon
I'm looking for an algorithm to find a set of largest inscribed rectangles of a concave polygon where each rectangle must be collinear with one of the edges of the polygon.
In other words, I want to ...
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The distribution of the shortest path through $n$ points
In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small?
More ...
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Smoothability of compact Alexandrov surfaces with curvature bounded from below
Let $(X,d)$ be compact metric space of curvature greater than $-1$ (in the sense of comparison triangles), assume that its Hausdorff dimension is $2$. Then a result of Perelman says that $X$ is a 2-...
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History of the triangle inequality
I am currently preparing a talk that revolves around the triangle inequality.
Because this inequality is so well-established, I do not want to (in my talk) belabor too much upon the importance it ...
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Gromov hyperbolicity constant vs. Gromov-Hausdorff distance to a tree
Let $X$ be a compact, geodesic metric space which is Gromov hyperbolic with a constant $\delta>0$. To fix scaling, let us also assume that $X$ has diameter $1$.
To fix a definition of Gromov ...
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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 $...
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Graph metric approximating Euclidean metric
I've been reading Wolfram's recent articles about graph/mesh/grid structures as an analogy for physical space, and it seems to me that there will be a problem getting the notion of distance to work ...
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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$...
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Perimeter-halving center of a convex shape
Let $P$ be a convex polygon (or any convex body in $\mathbb{R}^2$)
with perimeter of length $1$. Call a chord $c$ of $P$ perimeter-halving
if half the perimeter lies to one side of $c$
(and so half to ...
- 151k
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Solids with constant surface area during "erosion"
Imagine a drug, a pill that you swallow, which is designed to dissolve in your
stomach at a constant rate. It must be shaped such that the surface area
remains constant when the volume is "eroded" ...
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Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped?
If $M$ is an $n \times n$ matrix, $|\det(M)|$ is the volume of the $n$-dimensional
parallelepiped spanned by the column vectors of $M$.
...
- 151k
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2-layer tilings with a center-of-gravity constraint
I've encountered a tiling problem with a physical constraint that
might place it outside the literature on tiling.
"Tiling" is a bit of a misnomer; it is a special type of cover.
All the tiles are ...
- 151k
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The "grassmannian" of a simplicial complex
This question is mainly a reference request – I have a construction which seems natural, so I am quite convinced it should be standard, but I don't know what it is called.
Take an $n$ dimensional ...
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Characterizations of Euclidean space
I posted this question at math.stackexchange.com but didn't get an answer. Is it a dumb question, eventually?
There are three ways of characterizing the abstract Euclidean space $E^n$ that are quite ...
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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 ...
- 13.6k
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Where to find English translation of Pansu's paper from Ann. Math?
Where can I find English translation of the following paper?
P. Pansu,
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. (French. English summary) [Carnot-...
- 28k
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An integral identity involving cotangents and Bessel functions
Numerical experiments suggest that the following integral identity holds for Bessel functions of the first kind,
$$J_2(t) = 12 \int_0^{1/2}\mathrm{d}x\,\cot \pi x \int_0^x \mathrm{d}y\, \cot \pi y \, ...
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Ping-pong progress through a quincunx
A quincunx or
Galton board consists of
staggered pegs from which ping-pong balls bounce and eventually display
a binomial / normal distribution in catch-bins. I am wondering if the
downward progress ...
- 151k
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CAT(0) groups that does not act on CAT(0) cubical complex
CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...
- 1,598
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A question about tiling Hilbert Space
Let H be an infinite dimensional and separable Hilbert Space. Let e be a positive real number-which can be arbitrarily small. Does there exist a denumerably infinite set S of pairwise disjoint and ...
- 6,215
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Doubling space without Besicovitch covering theorem?
A metric space is doubling if any ball of radius $2R$ can be covered by $N$ balls of radius $R$ and $N$ is fixed once forever.
Is there an example of complete length-metric space which is doubling, ...
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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:
&...
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