## New answers tagged mg.metric-geometry

1
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### Binary codes with upper and lower bound on pairwise distance

Instead of $\{0,1\}^n$, you may take as your code space a subset $S\subseteq\{0,1\}^n$ of diameter $D$. This will guarantee that, whatever code you define in $S$, its codewords will be at distance $\...

2
votes

### Convex hull in CAT(0)

There is a counterexample if instead of the CAT(0) condition a weaker notion of non-positive curvature is considered. A bicombing on a metric space distinguishes for each pair of points a geodesic ...

1
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Accepted

### Connectedness of fibers of almost Riemannian submersions

In the definition of submersion you should assume that $d_xf\colon \textrm{T}_x\to \textrm{T}_{f(x)}$ is onto.
In this case there is an almost horizontal lift $\phi_x\colon\textrm{T}_{f(x)}\to\textrm{...

2
votes

### Does $\mathbb Z^n$ contain $A_n$?

The even sublattice of $\mathbb Z^n$ is the root lattice $D_n$ which does not contain $A_n$ for $n\geq 4$. ($D_2=A_1\oplus A_1$ and $D_3=A_3$ are generally omitted in lists of root-systems) This ...

4
votes

Accepted

### Does $\mathbb Z^n$ contain $A_n$?

No, for no $n \neq 3$ does $A_n$ embed in $\mathbb{Z}^n$ (which I assume has the standard diagonal intersection form).
Call $r_1, \dots, r_n$ the roots generating $A_n$ with $r_i r_{i+1} = -1$ and $...

1
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### Dimension of Alexandrov space which is homeomorphic to a manifold

If $n$ is defined, then the statement has already been proved in the paper by Burago, Gromov, and Perelman.
However, there might be no such $n$; in other words, the space has infinite dimension in ...

5
votes

Accepted

### Interpretation and validity of modified Heisenberg uncertainty principle in a metric context?

Let me try to answer the question "How does this logarithmic interpretation of the Heisenberg uncertainty principle compare with the conventional understanding". There are two issues that ...

2
votes

### To place copies of a planar convex region such that number of 'contacts' among them is maximized

The left shape below has $3$ contacts (circled)
"between pairs of units" and hull area $> 3$, while
the right shape has $2$ contacts and area $3$.
So minimizing the hull area does not ...

2
votes

### When "$(\varepsilon,\delta)$-geodesic" cannot be a loop?

Assembling my comments into a partial answer:
You may be interested in the theorem that "in the presence of negative curvature, local quasi-geodesics are in fact global quasi-geodesics". ...

2
votes

### "Almost geodesics" in Riemannian manifolds which cannot be loops

Assume $M$ is complete.
(It is easy to construct counterexamples in the noncomplete case.)
The inverse function theorem implies that the map $f$ is locally invertible;
that is, if $p\in f(x)$, then ...

1
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### To maximize the volume of the polyhedron resulting from perimeter-halvings of a convex polygonal region

This is only a remark about one polygonal region $S$, a square.
As mentioned in this posting,
I computed numerically the max volume shape that can be made from
any folding of a square. Here it is:
...

2
votes

### Embedding a countably infinite metric space in $\ell^2(\mathbb Z_+)$

When embedding $f_n:\{ x_1, \ldots , x_{n+1}\}\to\mathbb R^n\cong \ell^2(\{ 1, \ldots ,n\})$, we can assume that $f_n(x_j)\in \ell^2(\{ 1, \ldots , j-1\})$ (and $f_n(x_1)=0$), by giving $\ell^2$ a ...

1
vote

### Conic neighborhoods ⇔ polyhedral

Just to close the question:
a proof is written in our note "Local characterization of polyhedral spaces".

3
votes

### A triangle comparison in CAT(0) spaces

This is just a bit more elaboration on Anton's nice example, and subsequent comment. Here is the picture of the two triangles:
Note that $p'x'y'$ is obtained from $pxy$ by rotating the side $\...

3
votes

Accepted

### A triangle comparison in CAT(0) spaces

No, it is not true.
Suppose $X$ glued from two solid plane triangles $\blacktriangle p x \bar y$ and $\blacktriangle y x\bar y$, the angle at $y$ is obtuce and the total angle at $\bar y$ is reflex.
...

2
votes

Accepted

### Forming paper bags that can 'trap' 3D regions of max surface area

Yes, if $S$ is convex, then you can form what is called in
Geometric Folding Algorithms (p. 382)
a perimeter-halving as follows.
Pick any point $x$ on $\partial S$, and another point $y \in \partial S$...

3
votes

Accepted

### Characterization of convexity by connectedness of hyperplane sections

Suppose $n\geqslant 2$.
Let $S$ be the boundary of $\tfrac12$-neighborhood of the unit $n$-sphere in $\mathbb{R}^{2{\cdot}n+1}$.
(Note that $S$ is homeomorphic to $\mathbb{S}^n\times\mathbb{S}^n$.)
...

3
votes

### Integer-distance sets

There has been recent progress concerning finitary integer distance sets in the plane. Greenfeld, Iliopoulou, and Peluse prove that if $P\subset [-N,N]^2$ is an integer distance set not contained ...

0
votes

### Neusis constructions

baragar.faculty.unlv.edu/papers/TwiceNotch.pdf
Arthur Baragar proved the equivalence of neusis and conchoid-assisted constructions, and that all complex numbers constructible by neusis/conchoid lie on ...

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