## New answers tagged discrete-geometry

3
votes

Accepted

### Can the graph of a symmetric polytope have more symmetries than the polytope itself?

The answer is No, the graph of an arc-transitive polytope cannot have more symmetries than the polytope.
The polytope and its graph have the exact same symmetries!
In the article "Capturing ...

- 11k

2
votes

Accepted

### What properties are preserved by quasi-isometries

One can think of quasi-isometric spaces as spaces which look the same when seen from far away. Examples of properties preserved under quasi-isometries are for example Gromov-hyperbolicity (for ...

- 335

0
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### Discrete isoperimetric problems

You may find answers to some of your questions in the book "Regular Figures" by Laszló Fejes Tóth, published in 1964. Following is the review of the book, as it appeared in the ...

- 7,010

2
votes

Accepted

### Relation of MSTs in the Euclidean plane to Delaunay triangulations

It cannot be for any planar triangulation. Say we have a triangle with vertices $x, y, z$ and $xy$ is the longest edge.
Consider a run of Prim’s MST algorithm which at each step adds an edge to a ...

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0
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### Which pyramids fill space?

About the tetrahedra problem, I have found this reference/proof that they cannot be used to fill 3D space (since the proven upper bound is smaller than $100 \cdot (1-2.7 \cdot 10^{-25})\%$: https://en....

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1
vote

### Distance of average of points to center of minimum enclosing ball

Assume $d=2\cdot k$.
If $d$ is large, then the vertices of the cube such that $\|v\|=k$ lie very densely in $(d-1)$-dimensional sphere.
If $n$ is large, but not that large, then you may choose $n-1$ ...

- 39.4k

1
vote

### How much smaller is the Čech complex than the Vietoris-Rips complex?

I'm going to offer an answer mainly to get an idea off my brain and maybe someone will point out why this is incorrect. However, in my view, a lot of discussions about Čech Vs Vietoris-Rips seem to ...

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