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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Maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1). Proof? [closed]
How to prove that the maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1)?
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infinite dimensional CAT(0) groups
Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and ...
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Ricci curvature : beyond heat-like flows
Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs.
There are at least two versions of Ricci curvature in the ...
- 7,853
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Smallest regular simplex containing the unit cube in $R^n$
What is the length $e_n$ of the edge of the smallest $n$-dimensional regular simplex $S_n$ containing the $n$-dimensional unit cube $Q_n$?
In particular, is there $n$ such that $e_n<\sqrt{2}(n+1-\...
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Three squares in a rectangle
One of my colleagues gave me the following problem about 15 years ago:
Given three squares inside a 1 by 2 rectangle, with no two squares overlapping, prove that the sum of side lengths is at most 2. (...
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bi-Lipschitz gluing
Let $H$ be the Heisenberg group with
left invariant sub-Riemannian metric and $\varepsilon>0$ is small.
Let us denote by $|x-y|_H$ the distance from $x$ to $y$ in $H$.
I have a bi-Lipschitz ...
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Affine "real algebraic geometry" of hyperbolic space?
Real algebraic geometry, at least to start with, traditionally studies the zero-sets of real polynomials in a given set of variables. But treating, say, the Euclidean plane as an uncoordinatized ...
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Pythagorean theorem for right-corner hyperbolic simplices?
My answer to the "Favorite equations" question was the Pythagorean theorem for right-corner tetrahedra:
Euclidean: $A^2+B^2+C^2=D^2$
Hyperbolic: $\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}−\...
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680
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Open bilinear maps that are not uniformly open
A map $f\colon X\to Y$ between metric spaces is uniformly open whenever for each $\varepsilon >0$ there is $\delta >0$ such that for any $x\in X$ one has
$$B_Y\big(f(x),\delta\big)\subseteq f\...
- 11.3k
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What is the longest algebraic curve?
Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$.
Is $L(d)\leq d P(E)/2$, where $P(E)$ is the ...
- 7,836
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Ratio of circumscribed/inscribed $(n{-}1)$-gons
As a discrete analog of the MO question,
"Löwner-John Ellipsoid: incribed and circumscribed,"
I've been wondering what might be the maximum ratio
of this quantity?
Let $P$ be a convex ...
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Ping-pong relief map of a given function z=f(x,y)
I have an idea to design a type of
Galton's Board
to "draw" a relief map of a given two-dimensional function $z=f(x,y)$.
A typical Galton's Board drops, say, ping-pong balls through a series
of evenly ...
- 151k
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Closed geodesics avoiding points in hyperbolic surfaces
Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...
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Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
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Has Cheeger's 'de Rham cohomology' of metric measure spaces been studied beyond its definition?
In J. Cheeger's 'Differentiability of Lipschitz Functions on Metric Measure Spaces' (Geometric and Functional Analysis, 1999, Vol. 9 pp 428-517, see here), a 'de Rham cohomology group' $H_{dR}^1(Z,\mu)...
user14166
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Functions approximated by rolling epicycle curves
Imagine a decreasing sequence of (positive) radii $r_1 > r_2 > r_3 > \cdots$
and a series of nested circles $C_1 \supset C_2 \supset C_3 \supset \cdots$
with these radii,
initially each ...
- 151k
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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 ...
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Finite set of non-collinear points on plane with every point having ≥ 3 equidistant neighbors? [closed]
Does there exist a finite set of points on the Euclidean plane, such that:
No 3 points are collinear, and
Every one of the points has (at least) three other points in the set at the same distance ...
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Explanation for gamma function in formula for $n$-ball volume
It is well-known that the volume of the unit ball in n-space is $\pi^{n/2}/\Gamma(n/2+1)$. Do you know of a proof which explains this formula? Any proof which does not treat the cases $n$ even and $n$ ...
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Why are isometries of Minkowski space necessarily linear?
The Mazur-Ulam theorem says that any surjective isometry of normed vector spaces is affine. This argument doesn't seem to apply to Minkowski space (of special relativity) since the metric is ...
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Solid angles of a tetrahedron
This is a problem I have had for a while. For a triangle, the side opposite the largest angle has the largest length (and similarly for smallest angle). For a tetrahedron, the question is whether the ...
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How to smootly interpolate between möbius transformations?
If you have two Möbius transformations represented as:
$f(z) = \frac{az + b}{cz + d}$
$g(z) = \frac{pz + q}{rz + s}$
where $a, b, c, d, p, q, r, s, z \in \mathbb{C}$
Is it possible to derive a ...
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Shortest-path Distances Determining the Metric?
The metric of a Riemannian manifold determines the shortest
distance between any two points.
I assume the reverse holds? That is, if you are given the
shortest distance d(x,y) between every pair of ...
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The Symmetry of a Soccer Ball
Let $P$ be a polyhedron which satisfies the following three conditions:
$P$ is built out of regular hexagons and regular pentagons.
Three faces meet at each vertex.
$P$ is topologically a sphere.
An ...
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Database of integer edge lengths that can form tetrahedrons
Is there a collection of lists of six integer edge lengths that form a tetrahedron? Is there a computer program for generating such lists? I need to find approximately thirty such tetrahedral ...
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Cheap, non-constructive, free group generating rotations for Banach-Tarski
Stan Wagon's exposition of Banach-Tarski (for example) includes a beautiful explicit construction of two 2-sphere rotations which generate a free subgroup of the rotation group.
For teaching purposes ...
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Can a tangle of arcs interlock?
Can a (finite) collection of disjoint circle arcs in $\mathbb{R}^3$ be interlocked in the sense in that they cannot be separated, i.e. each moved arbitrarily far from one another while remaining ...
- 151k
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For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?
Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature?
An obvious case is when $Y$ ...
- 1,069
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Smallest containing simplex
Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$.
What is known about $V_n$? Is there a ...
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Steiner's inequality reference request
I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have
$$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$
where
$$\...
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Progress on Gromov's Conjecture of the bound of total Betti numbers
This question is a reference request.
Let $(M,g)$ be a Riemannian manifold of dimension $n$, and $b_i(M) = \dim H_i(M,\mathbb{R})$. Gromov proved it that there are constants $C(n)$ such that, if the ...
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Right triangle with edge lengths equal to regular unit polygon edge lengths
This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...
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When do two holonomy maps determine flat bundles that are isomorphic as just bundles (w/o regard to the flat connections)?
Suppose we have a surface S (although the question might make as much sense in higher dimensions) and a topological group G. The data of a flat vector bundle on S (up to isomorphism) is the same as a ...
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When does a pair of homotopic Lipschitz functions fail to admit a Lipschitz homotopy?
Let $(M,d)$ and $(N,\rho)$ be metric spaces. A function $f: M \to N$ is Lipschitz if there exists some constant $\kappa \geq 0$ so that $\rho(f(x),f(y))$ is smaller than $\kappa d(x,y)$ for all points ...
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Sets of evenly distributed points in the Euclidean plane
Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection
with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite?
If the answer is yes, can $P$...
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Quasi-isometry groups of metric spaces
Given a metric space $(X, d)$, we can consider the set of all quasi-isometries $f: X \to X$, and quotient out by the equivalence relation identifying $f$ and $g$ if $\sup_{x \in X}d(f(x), g(x))$ is ...
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Lattice points and convex bodies
Given are two convex bodies $K, L \subset \mathbb{R}^n$ that contain the origin as an interior point. Assume the number of integer points contained in $\lambda K$ equals the number of integer points ...
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Recognizing Lipschitz functions up to change of target metric
Let $K$ be a compact subset of $\mathbb{R}^n$ (for simplicity, I am happy to take $K=\overline{B(0,1)}$ for now if it is easier).
Let $f:K \rightarrow \mathbb{R}^m$ be a continuous function.
Is ...
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The space of triangles that fit inside a given triangle, parametrized by edge lengths
Given a triangle T with sides a, b, and c, describe its "fitting set," the set of all points (x,y,z) in 3-dimensions for which a triangle with sides x, y, z exists that fits in T.
Such a set lies in ...
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Category of data sets, motivated by persistent homology?
Is there a useful or agreed-upon category of data sets? In particular, I'm thinking about a point cloud and wondering what an acceptable morphism between point clouds "should" be.
Edit/Clarification:...
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Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?
Question 1. Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees?
Here an integral of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ ...
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How far is a set of vectors from being orthogonal?
Given some vectors, how many dimensions do you need to add (to their span) before you can find some mutually orthogonal vectors that project down to the original ones?
Or, more formally...
Suppose $...
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Double kissing problem
Consider two touching unit balls which will be called central balls. What is the maximum number $k$ of non-overlapping unit balls so that each ball touches as least one of two central balls?
An easy ...
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How metric is Riemannian geometry
Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by
$$d(x, y) = \text{inf} \int_a^b \| \dot\gamma(t) \| \, dt\,,$$...
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Conjugate points on cut locus
Let $M$ be a Riemannian with nonempty boundary $\partial M$.
Define multiplicity of $x\in M$ as the number of minimizing geodesics from $x$ to $\partial M$.
The following fact seems to be standard:
...
- 45k
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Geometric meaning of a trigonometric identity
It follows from the law of cosines that if $a,b,c$ are the lengths of the sides of a triangle with respective opposite angles $\alpha,\beta,\gamma$, then
$$
a^2+b^2+c^2 = 2ab\cos\gamma + 2ac\cos\beta +...
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Pushing convex bodies together
Given two convex bodies $A$ and $B$, in $\mathbb R^3$ let's say. We define $A(t)$ and $B(t)$ as $A+xt$ and $B+yt$ where $x,y$ are two arbitrary points. (That is the Minkowski sum, so the two bodies ...
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Are all well behaved "mean" functions on $\mathbb{R}^+$ equivalent?
Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
$M(M(a,b),M(a,c))=...
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Must a path of compact sets in $X$ descend to a path in $X$?
(I am most interested in the case $X=\mathbb R^2$, but of course one could ask the same question for manifolds, or metric spaces in general.)
Let $\text{Com}(\mathbb R^2)$ denote the space of nonempty ...
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