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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 …
Vidit Nanda's user avatar
  • 15.5k
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 …
Vidit Nanda's user avatar
  • 15.5k
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 …
Vidit Nanda's user avatar
  • 15.5k
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 …
Vidit Nanda's user avatar
  • 15.5k
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 …
Vidit Nanda's user avatar
  • 15.5k
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 …
Vidit Nanda's user avatar
  • 15.5k
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 …
Vidit Nanda's user avatar
  • 15.5k