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1 vote

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 $\...
aleph's user avatar
  • 503
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 ...
Giuliano Basso's user avatar
1 vote
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{...
Anton Petrunin's user avatar
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 ...
Roland Bacher's user avatar
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 $...
Marco Golla's user avatar
  • 10.3k
1 vote

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 ...
Anton Petrunin's user avatar
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 ...
Carlo Beenakker's user avatar
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 ...
Joseph O'Rourke's user avatar
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". ...
Sam Nead's user avatar
  • 25.3k
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 ...
Anton Petrunin's user avatar
1 vote

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: ...
Joseph O'Rourke's user avatar
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 ...
Christian Remling's user avatar
1 vote

Conic neighborhoods ⇔ polyhedral

Just to close the question: a proof is written in our note "Local characterization of polyhedral spaces".
Anton Petrunin's user avatar
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 $\...
Mohammad Ghomi's user avatar
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. ...
Anton Petrunin's user avatar
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 $y \in \partial S$...
Joseph O'Rourke's user avatar
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 $\mathbb{S}^n\times\mathbb{S}^n$.) ...
Anton Petrunin's user avatar
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 ...
Zach Hunter's user avatar
  • 3,335
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 ...
Michael Ejercito's user avatar

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