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Results tagged with discrete-geometry
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user 18263
Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.
15
votes
3
answers
2k
views
Combinatorial analogues of curvature
There appear to be many "combinatorial" definitions of curvature as applied to finite simplicial (or regular CW) complexes. For instance, we have the ideas of Cheeger, Muller and Schrader, of Forman a …
4
votes
0
answers
86
views
Efficient CW structures on squarefree semi-algebraic set
General Setup
Given a collection of $k$ polynomials (with real coefficients) in $n$ real variables, say $f_i(x_1,\ldots,x_n)$, let $V \subset \mathbb{R}^n$ correspond to those $x$-values for which ev …
5
votes
2
answers
561
views
Covering convex polygons with inscribed disks
The following problem came up when discussing mapping software (e.g., Google maps) with computer scientists. By $B(c,r)$ I mean the planar disk (open or closed, it doesn't matter) of radius $r$ around …
4
votes
Accepted
Discrete Morse function from smooth one
This is a rapidly developing area, and there are many short-cuts if all you want to do is compute the homology of sub-level sets of $f$. To answer your main question, as Liviu has already mentioned: t …
6
votes
Tetris-like falling sticky disks
Regarding the question
Has this process, or something close to it, been studied before?
I was recently made aware of an intriguing approach of Bob Macpherson and his post-doc Ben Schweinhart at …
13
votes
3
answers
832
views
What fraction of n-point sets in the unit ball have diameter smaller than 1?
This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean spa …
8
votes
Accepted
Combinatorial distance between simplicial complexes
This got too long for a comment, so I am placing it here.
I don't think there is a theory already out there, but that should not be too surprising. After all, the "combinatorial distance" between a s …