Questions tagged [mg.metric-geometry]

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

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73 votes
10 answers
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Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
Terry Tao's user avatar
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27 votes
8 answers
5k views

Representability of finite metric spaces

There have been a couple questions recently regarding metric spaces, which got me thinking a bit about representation theorems for finite metric spaces. Suppose $X$ is a set equipped with a metric $d$...
Matt Noonan's user avatar
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47 votes
0 answers
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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
40 votes
3 answers
15k views

Distributing points evenly on a sphere

I am looking for an algorithm to put $n$-points on a sphere, so that the minimum distance between any two points is as large as possible. I have found some related questions on stackoverflow but ...
CPJ's user avatar
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86 votes
4 answers
10k views

Is the sphere the only surface with circular projections? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are circular?

Several ancient arguments suggest a curved Earth, such as the observation that ships disappear mast-last over the horizon, and Eratosthenes' surprisingly accurate calculation of the size of the Earth ...
Joel David Hamkins's user avatar
69 votes
4 answers
11k views

$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$

I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem, a claim repeated in this ...
Joseph O'Rourke's user avatar
49 votes
4 answers
4k views

What fraction of the integer lattice can be seen from the origin?

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes ...
Joseph O'Rourke's user avatar
25 votes
7 answers
9k views

Uniformly Sampling from Convex Polytopes

How to choose a point uniformly from a convex polytope $P \subset [0,1]^n$ defined by some inequalities, $Ax < b$? (Here $A$ is an $m \times n$ matrix, $x \in \mathbb{R}^n$, and $b \in \mathbb{R}^...
john mangual's user avatar
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55 votes
6 answers
7k views

Is it possible to partition $\mathbb R^3$ into unit circles?

Is it possible to partition $\mathbb R^3$ into unit circles?
Zarathustra's user avatar
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44 votes
11 answers
25k views

Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
Xerxes's user avatar
  • 441
26 votes
2 answers
4k views

3D models of the unfoldings of the hypercube?

There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the tesseract, into 3D.1 These unfoldings (or "nets") are analogous to the 11 unfoldings of the 3D cube into the plane.2 ...
Joseph O'Rourke's user avatar
96 votes
7 answers
19k views

Can we cover the unit square by these rectangles?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1. It is easy to show that $$\sum_{1 \...
Kaveh's user avatar
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63 votes
6 answers
4k views

Shortest closed curve to inspect a sphere

Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and exterior to $S$ which has the property that every point $x$ on $S$ is visible to some point $y$ of $...
Joseph O'Rourke's user avatar
36 votes
10 answers
6k views

Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature

A curve in the plane is determined, up to orientation-preserving Euclidean motions, by its curvature function, $\kappa(s)$. Here is one of my favorite examples, from Alfred Gray's book, Modern ...
Joseph O'Rourke's user avatar
27 votes
6 answers
10k views

Almost orthogonal vectors

This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
Matthew Daws's user avatar
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10 votes
0 answers
725 views

Topological dimension, Hausdorff dimension, and Lipschitz mappings

I can prove the following result. Here $\operatorname{dim} X$ stands for the topological dimension and $\mathcal{H}^n$ denotes the Hausdorff measure. Theorem. Suppose that $f:\mathbb{R}^n\supset\...
Piotr Hajlasz's user avatar
10 votes
2 answers
886 views

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 $D \subseteq \mathbb{R}^2$...
Asaf Shachar's user avatar
  • 6,611
7 votes
2 answers
1k views

G-spaces and manifolds

In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms: The space is metric The space is finitely compact, i.e., a ...
Dror Atariah's user avatar
100 votes
6 answers
5k views

Light rays bouncing in twisted tubes

Imagine a smooth curve $c$ sweeping out a unit-radius disk that is orthogonal to the curve at every point. Call the result a tube. I want to restrict the radius of curvature of $c$ to be at most 1. I ...
Joseph O'Rourke's user avatar
99 votes
6 answers
11k views

Is there an analogue of curvature in algebraic geometry?

I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...
Paul Siegel's user avatar
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60 votes
1 answer
6k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...
Benjamin Dickman's user avatar
51 votes
3 answers
2k views

Can the sphere be partitioned into small congruent cells?

On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
Wlodek Kuperberg's user avatar
37 votes
1 answer
2k views

Sofa in a snaky 3D corridor

What is the largest volume object that can pass though a $1 \times 1 \times L$ "snaky" corridor, where $L$ is large enough to be irrelvant, say $L > 6$.           ...
Joseph O'Rourke's user avatar
34 votes
6 answers
8k views

Covering a unit ball with balls half the radius

This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks": How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of ...
Joseph O'Rourke's user avatar
30 votes
5 answers
15k views

How to check if a box fits in a box?

How could I calculate if a rectangular cuboid fits in an other rectangular cuboid, it may rotate or be placed in any way inside the bigger one. For example would, (650,220,55) fit in (590,290,160), ...
user115086's user avatar
27 votes
8 answers
5k views

Convex hull in CAT(0)

Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset. Is it true that convex hull of $K$ is compact? Comments: Convex hull of $K$ = intersection of all closed convex sets ...
Anton Petrunin's user avatar
24 votes
1 answer
7k views

Hanging a ball with string

What is the shortest length of string that suffices to hang a unit-radius ball $B$? This question is related to an earlier MO question, but I think different. Assume that the ball is frictionless. ...
Joseph O'Rourke's user avatar
23 votes
3 answers
2k views

Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$. Say that $S$ is an integer-distance set if every pair of points in $S$ is separated by an integer Euclidean distance. ...
Joseph O'Rourke's user avatar
17 votes
2 answers
5k views

Square of the distance function on a Riemannian manifold

Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function $$dist^2\colon M\times M\to \mathbb{R}$$ given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this ...
asv's user avatar
  • 21.1k
14 votes
2 answers
2k views

Surface area of an $\ell_p$ unit ball?

Are there any known formulas or approximations for the surface area of a unit ball in $d$ dimensions under the $\ell_p$ norm? As obvious examples, it is of course well-known that the surface area of ...
Gene's user avatar
  • 141
14 votes
1 answer
2k views

Dao's theorem on six circumcenters associated with a cyclic hexagon

This questions from Ngo Quang Duong's paper In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...
Oai Thanh Đào's user avatar
13 votes
5 answers
1k views

Packing obtuse vectors in $\mathbb{R}^d$

I came across this attractive theorem: Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that form an obtuse angle with one another. This was proved1 as a corollary of a lemma about ...
Joseph O'Rourke's user avatar
13 votes
3 answers
1k views

Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard.     ...
Joseph O'Rourke's user avatar
12 votes
5 answers
6k views

Subset of the plane that intersects every line exactly twice

In a comment to this question, Tim Gowers remarked that using the axiom of choice, one can show that there exists a subset of the plane that intersects every line exactly twice (although it has yet to ...
Tony Huynh's user avatar
  • 31.5k
12 votes
2 answers
927 views

Set of points with a unique closest point in a compact set

Let $K\subset\mathbb{R}^n$ be any compact set. Let $\operatorname{Unp}(K)$ be the set of points in $$ \operatorname{Unp}(K)=\{x\in\mathbb{R}^n\setminus K:\, \exists ! y\in K \ \ |x-y|=d(x,K)\}. $$ ...
Piotr Hajlasz's user avatar
10 votes
3 answers
4k views

Left invariant metric on ${\rm SL}_n(\mathbb{R})$

I am looking for a left invariant metric on $SL_n(\mathbb{R})$. If this is not possible, it would be acceptable to have a metric on $SL_n(\mathbb{R})/SO_n(\mathbb{R})$ or something like that. Is there ...
safsaf32's user avatar
  • 109
10 votes
1 answer
1k views

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 ...
Hans-Peter Stricker's user avatar
9 votes
1 answer
997 views

A chain of six circles associated with a conic

I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems: A chain of six circles ...
Oai Thanh Đào's user avatar
8 votes
4 answers
426 views

Inside-out polygonal dissections

A dissection of a polygon $P$ is a partition of $P$ into a finite number of pieces, which can then be rearranged (via planar translations and rotations) and joined (without overlap) to form a new ...
Joseph O'Rourke's user avatar
2 votes
1 answer
236 views

Smallest 3-ellipses that contain triangles

Reference: https://en.wikipedia.org/wiki/N-ellipse Question: How does one find and characterize the smallest 3-ellipses (n-ellipses with n =3) that contain a given triangle? 'Smallest' can mean 'least ...
Nandakumar R's user avatar
  • 5,401
63 votes
8 answers
14k views

Fair but irregular polyhedral dice

I am interested in determining a collection of geometric conditions that will guarantee that a convex polyhedron of $n$ faces is a fair die in the sense that, upon random rolling, it has an equal $1/n$...
Joseph O'Rourke's user avatar
60 votes
7 answers
25k views

Is the Jaccard distance a distance?

Wikipedia defines the Jaccard distance between sets A and B as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a book claiming that this is a metric. However, I couldn't find any ...
rgrig's user avatar
  • 1,335
60 votes
2 answers
4k views

Does this geometry theorem have a name?

Start with a circle and draw two tangent circles inside. The (black) inner tangent lines to the smaller circles intersect the large circle. The (red) lines through these intersection points are ...
Simon's user avatar
  • 509
57 votes
14 answers
18k views

Open problems in Euclidean geometry?

What are some (research level) open problems in Euclidean geometry ? (Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet) I should clarify a bit ...
43 votes
12 answers
2k views

Can a discrete set of the plane of uniform density intersect all large triangles?

Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximately equal to the area of the disc. Does the complement of S necessarily contain triangles ...
Roland Bacher's user avatar
39 votes
5 answers
3k views

Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points ...
Joseph O'Rourke's user avatar
35 votes
5 answers
3k views

Tiling the plane with incongruent isosceles triangles

It is not difficult to tile the plane with incongruent triangles. One could tile with equilateral triangles, and then partition each equilateral into three triangles, displacing their common ...
Joseph O'Rourke's user avatar
34 votes
6 answers
6k views

How to explain the concentration-of-measure phenomenon intuitively?

One way to phrase the "concentration-of-measure" phenomenon is that, for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$, "most of the mass is close to the equator, for any equator."1 Q. ...
Joseph O'Rourke's user avatar
27 votes
1 answer
1k views

Terrible tilers for covering the plane

Let $C$ be a convex shape in the plane. Your task is to cover the plane with copies of $C$, each under any rigid motion. My question is essentially: What is the worst $C$, the shape that forces the ...
Joseph O'Rourke's user avatar
27 votes
3 answers
12k views

Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Recently Mark McClure constructed and displayed the 261 unfoldings of the hypercube (tesseract) in response to the question, "3D models of the unfoldings of the hypercube?": The first 9 unfoldings ...
Joseph O'Rourke's user avatar

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