Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,406 questions
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Hearing the 17 planar symmetry groups
Though I'm sure it's not really hard to work out for myself, does anyone know a reference for the spectra of the Laplacian on the 17 flat compact orbifolds that underlie the 17 planar symmetry groups. ...
- 17.6k
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2
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Approximating a convex function by a piecewise linear function
Suppose I have a Lipschitz-continuous convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. I wish to approximate it on the unit ball by a piecewise-linear function $g:\mathbb{R}^n\rightarrow \...
- 161
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3-piece dissection of square to equilateral triangle?
At a workshop it was suggested that it likely remains an open problem
whether or not there is a 3- or 2 -piece
dissection
of a square to an equilateral triangle.
Can anyone confirm that this is ...
- 151k
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4
answers
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Inverse limit in metric geometry
Question. Did you ever see inverse limits to be used (or even seriousely considered) anywhere in metric geometry (but NOT in topology)?
The definition of inverse limit for metric spaces is given ...
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2
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Lipschitz constant for map between triangles
Let $T_1$ and $T_2$ be any two euclidean triangles with labeled sides. The sides are labeled respectively $e_1^1,e_2^1,e_3^1$ and $e_1^2,e_2^2,e_3^2$. Call $A:T_1\rightarrow T_2$ the affine map which ...
user99087
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1
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Status of Larry Guth's Sponge Problem
[Edited Jan 23, 2021]
Let $D^n$ be the $n$-dimensional unit radius disk in euclidean $\mathbb{R}^n$.
Larry Guth's Sponge Problem asks: Does there exist a constant $\epsilon=\epsilon_n$ such that every ...
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Can a shape rolling inside itself reproduce that shape?
Q. Is the circle the only shape that, when rolling inside itself,
has a point that draws out a scaled copy of itself?
Let $C$ be a simple, closed, smooth curve in the plane.
(Likely "smooth" can be ...
- 151k
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2
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Metric on one-point compactification
Is there a standard construction of a metric on one-point compactification of a proper metric space?
Comments:
A metric space is proper if all bounded closed sets are compact.
Standard means found in ...
- 45k
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0
answers
391
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Is "Escherian metamorphosis" always possible?
$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
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An open problem in convex geometry
Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
- 13.5k
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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 ...
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Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces
Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm.
Question: Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so ...
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3
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A metric for Grassmannians
I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (https://doi.org/10.1007/BF02585431). I'm having a technical problem. Sorry for ...
user11178
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4
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More than $n$ approximately orthonormal vectors in $R^n$
This question was asked at math.stackexchange, where it got several upvotes but no answers.
It is impossible to find $n+1$ mutually orthonormal vectors in $R^n$.
However, it is well established that ...
- 1,160
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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 $\...
- 151k
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orientations for zero-dimensional manifolds
I am teaching a course on manifolds, and soon I will have to prove the Stokes' theorem which, of course, involves defining oriented manifolds. There are many ways to define an oriented manifold. My ...
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2
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Partitioning a Rectangle into Congruent Isosceles Triangles
Is it possible to partition any rectangle into congruent isosceles triangles?
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3
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Stronger version of the isoperimetric inequality
I have been searching for a version of the isoperimetric inequality which is something like:
$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r_{out}^2 - r_{in}^2)$ where $r_{out}$ and $r_{in}$ ...
- 2,641
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3
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About MF Atiyah and R Bott's 1983 paper
I am a theoretical physics major student working on string theory. I want to understand the work of MF Atiyah and R Bott, "The Yang-Mills equations over riemann surfaces" . What kinds of mathematical ...
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3
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Use of n-transitivity in finite group theory
Hello, apparently finite groups which are n-transitive with n>5 are only the permutation groups Sn or the alternating groups An+2, see e.g. page 226 this book by Isaacs http://books.google.fr/books?...
- 2,901
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4
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Orthogonal mud cracks and Maxwell's reciprocal figures
Is there a succinct mathematical/physical explanation of why mud cracks
tend to meet orthogonally?
Wikipedia image in this ...
- 151k
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6
answers
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Thales' semicircle theorem in higher dimensions
Thales semicircle theorem says that an angle inscribed in a semicircle is a right angle.
Q1. Does a cone with apex on a hemisphere and encompassing the circular base
have a solid angle ...
- 151k
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3
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$n$-dimensional Voronoi diagram
I need to compute the Voronoi diagram of a set of points in $R^n$.
I'm quite unschooled on the topic, could someone point me to the right references so that I can
a) understand the theory behind it;
b)...
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4
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815
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Unlinked interlocking planar polygons
Let $P$ and $Q$ be the boundary segments of two planar simple polygons.
View these boundaries as rigid wires.
Fix $Q$ in, say, the $xy$-plane, and imagine $P$ arranged in $\mathbb{R}^3$ so that $P$ ...
- 151k
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votes
1
answer
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Concurrent normals conjecture
It is conjectured that if $K$ is a convex body in $n$-dimensional Euclidean space, then there exists a point in the interior of $K$ which is the point of concurrency of normals from $2n$ points on the ...
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votes
1
answer
616
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Acute triangles in "obtuse" polygons?
Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute?
I conjecture ...
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2
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Do two new special points in any triangle exist?
There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.
Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $...
- 1,776
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covering a square with unit squares
Can some square of side length greater than $n$ be covered by $n^2+1$ unit squares? (The unit squares may be rotated. The large square and its interior must be covered.)
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Continuity in terms of lines
Let $f: \mathbb R^n \rightarrow \mathbb R^n$, where $n> 1$ be a bijective map such that the image of every line is a line.
Is $f$ continuous?
I think it is, but the proof isn't immediately ...
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2
answers
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How to characterize the regularity of a polygon?
In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
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A cube is placed inside another cube
I known following problem with two square
$$Area(1)+Area(3)=Area(2)+Area(4).$$
My question. Is this problem true for two cubes?
We place a cube $XYZT.X'Y'Z'T$ into another cube $ABCD.A'B'C'D',$ ...
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Lattice n-gons with ordered side lengths 1,2,3,...,n
Consider the octagon in the Cartesian plane with vertices at (0,0), (1,0), (1,2), (4,2), (4,6), (7,2), (7,8), and (0,8).
Are there other (infinitely many) polygons, such as this, lying entirely in the ...
- 3,707
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1
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The 4th vertex of a triangle?
I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...
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1
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Gromov's articles suitable for master students
I'm a master student and I have read "Monotonicity of the volume of intersection of balls" by Gromov and it was a great experience. When trying to fill the gaps, I often end up finding some ...
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2
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What is known about Ulam's problem of metric spaces with isometric squares?
Background
In the book Problems in Modern Mathematics, S. Ulam asks the following question:
Suppose $A$ and $B$ are metric spaces, such that $A^2$ and $B^2$ equipped with the 2-metric $d((x_1, y_1),(...
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3
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Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.
Is the following fact true?
Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \...
- 1,284
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5
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Diameter of universal cover
Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric).
What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or $\pi_1(...
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2
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1k
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When is a bi-Lipschitz homeomorphism smoothable?
Suppose I have a smooth Riemannian manifold $X$ with induced distance function $d$, and a bi-Lipschitz (with respect to $d$) homeomorphism
$$\phi: X \to X.$$
Under what circumstances could $\phi$ be ...
- 1,601
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571
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Spearing rolling hula hoops
Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, d-\...
- 151k
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Characterization of a sphere: every "sub-sphere" has two centers
Let me ask this question without too much formalization:
Suppose a smooth surface $M$ has the property that for all spheres $S(p,R)$ (i.e. the set of all points which lie a distance $R\geq 0$ from $p ...
- 1,461
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votes
2
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758
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(Non)existence of mirrors with more than two foci
Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...
- 1,461
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votes
1
answer
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Random walk on a Penrose tiling
Pólya proved that a random walk on $\mathbb{Z}^2$ almost surely returns to the
origin, or, equivalently, returns to the origin infinitely often.
It was subsequently established that in $\mathbb{Z}^3$, ...
- 151k
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3
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734
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Embedding expanders in CAT(0) spaces
It is well-known that expanders are hard to embed into Hilbert (or $\ell^p$) spaces - any embedding of an expander with $n$ vertices has distortion $\Omega(\log n)$.
Can anyone provide a reference (...
- 2,449
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2
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1k
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Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?
I am interested in the Hausdorff dimension of the Apollonian circle packing.
There seem to be two numerical calculations of the value:
1.305686729(10)
from P.B ...
- 10.7k
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1
answer
530
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Dividing a polyhedron into two similar copies
The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
15
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2
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1k
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Is every connected metrizable locally path connected space a length space?
Does every connected metrizable locally path connected topological space $X$ admit a compatible metric $d$ so that $(X,d)$ is a length space?
(Edit to correct definition: Recall that a metric space $(...
- 1,968
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3
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1k
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Optimal inspection path on a sphere
Suppose you would like to "inspect" every point of a unit-radius
sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$
on $S$, but you can only see a distance $d$ from where you ...
- 151k
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2
answers
737
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Tiling survey that updates "Tilings and patterns"?
Can anyone suggest a survey (or surveys) that provides an update to Tilings and patterns by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one.
I am ...
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1
answer
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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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