Questions tagged [mg.metric-geometry]

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

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Concerning proofs from the axiom of choice that ℝ³ admits surprising geometrical decompositions: Can we prove there is no Borel decomposition?

This question follows up on a comment I made on Joseph O'Rourke's recent question, one of several questions here on mathoverflow concerning surprising geometric partitions of space using the axiom of ...
Joel David Hamkins's user avatar
42 votes
0 answers
791 views

A kaleidoscopic coloring of the plane

Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
Taras Banakh's user avatar
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32 votes
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906 views

Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$? There are some results of the ...
Mikhail Ostrovskii's user avatar
31 votes
2 answers
2k views

Tiling of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes, that is, it is a curve of constant width. (The width between two parallel tangents to the curve are independent of the ...
Per Alexandersson's user avatar
30 votes
0 answers
736 views

Is there an Ehrhart polynomial for Gaussian integers

Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
David E Speyer's user avatar
28 votes
0 answers
537 views

Can every 3-dimensional convex body be trapped in a tetrahedral cage?

Can every 3-dimensional convex body be trapped in a tetrahedral cage? Although the question is fairly unambiguous, I give all relevant definitions: $\bullet$ A subset $C$ of $\mathbb{R}^n$ is an $n$-...
Wlodek Kuperberg's user avatar
28 votes
0 answers
809 views

Blocking light with mirrored convex objects

There is a long-unsolved problem posed by Janos Pach, sometimes known as the enchanted forest problem, which asks if it is possible to block a point light source in the plane from reaching infinity by ...
Joseph O'Rourke's user avatar
25 votes
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1k views

Curves on potatoes

On twitter recently, Robin Houston brought up this problem from a mathematical puzzle book of Peter Winkler: The puzzle is attributed to the book "The mathemagician and the pied piper", and ...
Ian Agol's user avatar
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25 votes
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343 views

Can 4-space be partitioned into Klein bottles?

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles, or into disjoint unit circles, or into congruent copies of a real-analytic curve (Is it possible to partition $\mathbb R^3$ ...
Joseph O'Rourke's user avatar
22 votes
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377 views

What is the covering density of a very thin annulus? Is it $\frac{\pi\sqrt{51\sqrt{17}-107}}{16}$?

Take some very small $\epsilon>0$, and consider the annulus/ring given by the set $\{(r,\theta)\ |\ 1-\epsilon\le r\le1\}\subset \mathbb{R}^2$. We wish to place translated copies of this annulus ...
RavenclawPrefect's user avatar
22 votes
0 answers
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Boundaries of noncompact contractible manifolds

It is known that a manifold $B$ bounds a compact contractible topological manifold if and only if $B$ is a homology sphere. The "only if" direction follows by excising a small ball in the interior of ...
Igor Belegradek's user avatar
22 votes
0 answers
1k views

Random Distance Matrices

My question is motivated by the following recent paper: http://arxiv.org/abs/1110.6333 Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...
ght's user avatar
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19 votes
0 answers
830 views

I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable?

Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map: $f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$ with the following ...
Malkoun's user avatar
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19 votes
0 answers
553 views

"Japanese Theorem" on cyclic polygons: Higher-dimensional generalizations?

A beautiful theorem known as the Japanese Theorem (Wikipedia, MathWorld) says that, no matter how one triangulates a cyclic (inscribed in a circle) polygon, the sum of the radii of the incircles is ...
Joseph O'Rourke's user avatar
18 votes
0 answers
473 views

Trapping lightrays with segment mirrors

Q. Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors? I posed this question in several forums before (e.g., here and in an ...
Joseph O'Rourke's user avatar
18 votes
0 answers
660 views

The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in a unit $d$-dimensional cube, with perfectly elastic wall collisions. Let $k=n^{\frac{1}{d}}$. For example, in 3D, $d=3$, with $n=...
Joseph O'Rourke's user avatar
18 votes
0 answers
501 views

Lipschitz constant of a homotopy

Let $n>0$, $\mathbb S^n$ be $n$-sphere and $1\in \mathbb S^n$ be its north pole. A am looking for an example of compact manifold $M$ with a continuous $n$-parameter family of maps $h_x\colon M\to ...
Anton Petrunin's user avatar
17 votes
0 answers
361 views

Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?

Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation it stays a convex polytope, it stays a combinatorial dodecahedron (i.e. its ...
M. Winter's user avatar
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17 votes
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Almost monochromatic point sets

There are many sort of equivalent theorems about monochromatic configurations in finite colorings, such as Van der Waerden, Hales-Jewett or Gallai's theorem, the latter of which states that in a ...
domotorp's user avatar
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17 votes
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220 views

GPS calculations under $L^p$ norms

GPS calculations require finding a sphere externally tangent to four given spheres, an Apollonian problem in $\mathbb{R}^3$. The center of that fifth sphere is one of the $16$ possible solutions to ...
Joseph O'Rourke's user avatar
17 votes
0 answers
474 views

Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to (0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set $\{x_i\...
Mikhail Ostrovskii's user avatar
17 votes
0 answers
725 views

Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
Kevin Johnson's user avatar
16 votes
0 answers
560 views

Snakes on a plane

A sleeping bag for a baby snake in $d$ dimensions (no, really) is a subset of $\mathbb{R}^d$ which can cover (via translation and rotation) every (piecewise-smooth for concreteness) curve of unit ...
Noah Schweber's user avatar
16 votes
0 answers
388 views

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 ...
Noah Schweber's user avatar
16 votes
0 answers
2k views

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? ...
alvarezpaiva's user avatar
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16 votes
0 answers
289 views

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 ...
Misha's user avatar
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16 votes
0 answers
751 views

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 ...
Bill Johnson's user avatar
15 votes
0 answers
380 views

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)...
user avatar
15 votes
0 answers
516 views

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 ...
Joseph O'Rourke's user avatar
15 votes
0 answers
746 views

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 ...
HenrikRüping's user avatar
14 votes
0 answers
367 views

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 ...
Dan's user avatar
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14 votes
0 answers
259 views

Regular $n$-gon with diagonals: bounds on area of largest cell?

Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet). I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...
Dan's user avatar
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14 votes
0 answers
201 views

Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?

A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article: W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$. In Convex ...
Yemon Choi's user avatar
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14 votes
0 answers
477 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one: Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
Piotr Shatalin's user avatar
14 votes
0 answers
548 views

Who conjectured that a transitive projective plane is Desarguesian?

The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved. ...
Gordon Royle's user avatar
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13 votes
0 answers
638 views

Covering number estimates for Hölder balls

Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
ABIM's user avatar
  • 5,019
13 votes
0 answers
464 views

What are the known convex polyhedra with congruent faces?

Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
RavenclawPrefect's user avatar
13 votes
0 answers
252 views

Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$

An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane. The following questions are motivated by Anton Petrunin's Disc bounded ...
Wlodek Kuperberg's user avatar
13 votes
0 answers
488 views

Unit ball of smallest volume in a Hilbert geometry

In a letter to Felix Klein published in Mathematische Annalen 1895 (see here), Hilbert generalized the Cayley-Klein model of hyperbolic geometry by defining a metric on the interior of a convex body ...
alvarezpaiva's user avatar
  • 13.2k
13 votes
0 answers
562 views

Multiplicity of ball covering

Background. My questions are motivated by the following: A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete and Computational Geometry, 8 (1992) 109-130) considered the ...
Misha's user avatar
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13 votes
0 answers
251 views

Does there always exist a self dual polytope that contains a given polytope contained in its dual?

Suppose a polytope $P$ is contained in its dual polytope $\tilde{P}$. Does there always exist a polytope $Q$ that contains $P$ and is self dual $Q=\tilde{Q}$? Is there any bound on the minimal number ...
Huangjun Zhu's user avatar
13 votes
0 answers
733 views

$\epsilon$-nets with respect to the cut norm

The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in J}a_{i,j}\...
Aaron's user avatar
  • 794
12 votes
0 answers
1k views

The status of the journal “Forum Geometricorum”

The online journal Forum Geometricorum is a sort of central organ of elementary geometry (mainly triangle geometry and related topics). It has been published regularly since 2000 but seems to have ...
user131781's user avatar
  • 2,442
11 votes
0 answers
308 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
11 votes
0 answers
479 views

Are there 100 points that are part of every half-density part of the plane?

Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$? I am deliberately being vague ...
domotorp's user avatar
  • 18.3k
11 votes
0 answers
655 views

John-type theorems: trading structure for accuracy?

Given two symmetric convex bodies $B, B'$ in ${\bf R}^d$, define the Banach-Mazur distance $d(B,B')$ between them to be the least constant $\tau \geq 1$ such that $$ B \subset TB' \subset \tau B$$ for ...
Terry Tao's user avatar
  • 108k
11 votes
0 answers
210 views

Can a billiard rack be a square, for every number of balls?

A billiard rack is a rack, usually a triangle, that can hold a certain number of equal size billiard balls, such that the balls' centres cannot move within the rack. Can the rack be a square, for ...
Dan's user avatar
  • 2,341
11 votes
0 answers
327 views

Lattices and stable homotopy groups of spheres

The number $65520$ arises in two very different scenarios: It occurs in the formula for the theta series of the Leech lattice: $$ \Theta_{\Lambda_{24}}(q) = 1 + \sum\limits_{m=1}^{\infty} \dfrac{...
Adam P. Goucher's user avatar
11 votes
0 answers
292 views

Entropy, magnitude, diversity of finite metric spaces in number theory

I was reading the article by Tom Leinster, (Maximizing diversity in biology and beyond, arXiv link), and find it very interesting. Since I was searching for entropies of finite metric spaces I found ...
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