40

There is a wide body of work on this in connection with the classic De Bruijn–Erdős theorem.
De Bruijn–Erdős Theorem. Every set of $n$ points in the
plane (not all lying on the same line) determine at least $n$ lines.
There is a beautiful conjecture of Chen and Chvátal that the De Bruijn–Erdős theorem actually holds in every metric space, where lines ...

16

I'm reminded of W.A. Coppel's book which looks at these kinds of structures from a slightly different vantage point, namely closure systems. I can't actually find the book right now, but here's a quick synopsis of what's going on.
Given a set $X$ together with an arbitrary function $[\bullet,\bullet] : X^2 \rightarrow \mathcal{P}(X),$ let's call $A \...

8

The parametrization for the Klein bottle provided by Will Brian is
\begin{align}
x &= (a+b\cos v)\cos u\\
y &= (a+b\cos v)\sin u\\
z &= (b\sin v)\cos(u/2)\\
t &= (b\sin v)\sin(u/2)\\
\end{align}
where $a>b>0$.
This leads to the defining conditions:
\begin{align}
4a^2(x^2+y^2)&=(a^2-b^2+t^2+x^2+y^2+z^2)^2\\
y(z^2-t^2)&=2txz\\
...

8

Permit me to include this nice image from Day & Li
to illustrate @Igor's point that "in general on a surface, it [the cut locus] is a graph not a tree."
The source point $p$ is on the cat's forehead,
the other side in this rear-view.
Dey, Tamal K., and Kuiyu Li. "Cut locus and topology from surface point data." In ...

8

This is more of an extended remark than a full answer.
My message is:
The property $[x,x]=\{x\}$ does not necessarily hold, but you can always make it hold by taking a natural quotient.
Your axioms don't imply $[x,x]=\{x\}$, so you have to add it if you want it.
A pseudometric (e.g. a seminorm) will satisfy your axioms, and $[x,x]$ is the set of points ...

7

Yes, such examples do live. The following answer was given in "Metric spaces of non-positive curvature" by Bridson and Haefliger; thanks to GGT and Moishe Kohan.

7

If embedding means smooth embedding, the non-orientability of the Klein bottle $K$ implies that its stable normal bundle will also not be trivial. This implies that $K$ cannot be smoothly described as the solutions to an equation $f(\bf x)=\bf b$, with $\bf b \in \mathbb R^2$ a regular value of a smooth function $f: \mathbb R^4 \rightarrow \mathbb R^2$, i.e....

5

Consider a torus with three handles, where one handle is much larger than the others, and with a smooth and free $\mathbb{Z}_3$ action which permutes the handles.
Let $\gamma$ be a small geodesic loop going through one of the small handles. Let $z\gamma$ be an image of $\gamma$ under the group action which goes through the large handle. Let $p$ be a point ...

4

Step by step. First, we have the unit circle in the complex plane:
$$ S\ :=\ \{s\in\Bbb C:\ |s|=1 \} $$
We may even have a larger circle:
$$ C\ :=\ \{5\!\cdot\! s:\ s\in S\} $$
Then, a torus $\ T\ $ is a surface around circle
$\ C\times\{0\}\subseteq \Bbb C\times\Bbb R:$
$$ T\ :=\ \{ ((5+\Re s)\cdot c,\, \Im s)\,:\,\ (c\ s)\in S\times S\}
\quad\...

4

This paper https://jeb.biologists.org/content/221/19/jeb178988 contains an experimental investigation of egg rolling. Theoretically, it seems "the relationship of egg shape to egg movement (e.g. rolling) is an understudied topic".

3

This is not an answer, just some sketchy thoughts that are too long for the comment box. I and some other HHS enthusiasts are very interested in this question being answered; we've tried a fair bit and have set it aside, so I don't think they'll mind me trying to recall what some of the strategies and issues are.
It's indeed open for cocompact special ...

2

The most obvious example:
Suppose $U$ is a closed ball of radius $1$ in $\mathbb R^d$, and $T$ is the corresponding sphere. Then if $r = 1$, $N_r(U) = 1$ but $N_r(T) > 1$.
EDIT:
Let $X$ be any metric space such that there are three points $a,b, c$ with $d(a,b) \le d(a,c) < d(b,c)$. Then take $d(a,c) \le r < d(b,c)$, $T = \{b,c\}$ and $U = \{a,b,...

2

I'll discuss separately the question you asked and the exercise in the notes you linked to.
The question you asked is trivial as long as $k\geq3$. Just let $f$ be a projection onto the $3$-dimensional subspace spanned by $x,y,z$. You don't even need $\epsilon$, since $x,y,z$, and the angle are preserved exactly. (In fact, even if $k=2$, you can preserve the ...

2

Here's a compact example without boundary (a 2-torus).
Choose a topological disc $D$ on the two-torus $T$, and choose a Riemannian metric so that $D$ has very close Gromov-Hausdorff distance (say $\le 1$) to a segment of length $20$, while $T\smallsetminus D$ has diameter $\le 1$ (so metrically $D$ is predominant, while all the homotopic part lies in $T\...

2

Yes, it has to be a subgroup. Fix $v\in M$. We need to prove that $-v\in M$. It is sufficient to find an element of $M$ arbitrarily close to $-v$.
Choose $u_1,\ldots,u_n\in \mathbb{R}^n$ so that $v,u_1,\ldots,u_n$ are the vertices of a regular simplex with center at the origin. Choose large $N$ and consider the points $w_i\in M$ such that $\|Nu_i-w_i\|\...

2

Following up on Gerry Myerson's suggestion, the following graph from Ágoston and Pálvölgyi's Improved constant factor for the unit distance
problem may be a counterexample.
This is the $n=13$ example illustrating the maximum number (30) of unit distances among 13 points in the plane. Schade's 1993 thesis establishes that this is the unique maximal "unit ...

1

Depends on what you mean by equally spaced.
If you want that all points have pairwise identical distances, then this is not possible. This is only possible for at most $n+1$ points, but not more.
If you want to maximize the minimal distance of $n+2$ points in the $(n-1)$-sphere, you have to pick some of the vertices of the $n$-dimensional crosspolytope, ...

1

The $\epsilon$-covering number of the euclidean unit ball in $\mathbb R^d$ scales like $(1/\epsilon)^d$. More formally,
Lemma. If $\epsilon < 1$, then $(1/\epsilon)^d \le N(\epsilon, B_2) \le (3/\epsilon)^d$. Else $N(\epsilon,B_2) = 1$.
Proof. See Theorem 4.2 and Example 14.1 of this manuscript http://www.stat.yale.edu/~yw562/teaching/598/lec14.pdf. $\...

1

$\bigg\{ (z_1,z_2)\in \mathbb{C}^2 | |z_1|^2+|z_2|^2=1\bigg\}$ is
3-dimensional sphere $S$. When $ \Sigma =\bigg\{ (z_1,z_2)\in
\mathbb{C}^2 | |z_1| = |z_2| = \frac{1}{\sqrt{2}} \bigg\}$ is a torus in
$S$, then we know that $S$ is a union of two solid torus.
Here $((R+a\cos\ t)\cos\ \theta,(R+a\cos\ t)\sin\ \theta,a\sin\ t)$
is a parametrization for torus $...

1

As Benoit mentioned in one of his answers, ultrametric spaces are a class of metric spaces that can be analyzed, and there is a paper recently put on the arxiv, https://arxiv.org/abs/2003.10239, that carefully investigates when ultrametric spaces have finite metric dimension.
It also cites a considerable amount of literature where this concept appears in ...

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