All Questions
8 questions
15
votes
2
answers
780
views
How to characterize the regularity of a polygon?
In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
- 161
6
votes
0
answers
74
views
Roundest polyhedra: how well can we bound the edge count of their faces?
By "roundest" I mean having the lowest surface area for the highest volume, given a fixed number of faces $n$. There've been a few questions about them on here (including from me), but I'm ...
- 3,612
1
vote
2
answers
196
views
Partitioning unit square with equal frequency rectangles
If I had to partition the unit square $[0,1]\times[0,1]$ into $k^2$ rectangles such that the sum of their diagonals is minimum possible, I would simply choose the $k \times k$ grid of squares. Now ...
- 153
2
votes
1
answer
292
views
Facility location on manifolds
Facility location studies optimal placement of a certain number $n$ of points (facilities) in some region $R$. (https://en.wikipedia.org/wiki/Facility_location_problem)
The minimax facility location ...
- 5,979
1
vote
0
answers
48
views
Deployment and dispersion in triangular regions
Definitions (from C. Stanley Ogilvy's 'Tomorrow's Math'):
Deployment: To place a specified number $n$ of points (stations) in a region such that the maximum distance of any point in the region from ...
- 5,979
2
votes
0
answers
131
views
Optimal way to group points in the plane into clusters
Consider a strictly decreasing sequence $d = (d_k)_{k\ge 1}$ of distances in $(0,1)$. Given a constant $C>2$, we say that $d$ has the $C$-grouping property if any finite non-empty subset $S$ (of ...
- 1,761
7
votes
1
answer
318
views
Finding a short path using $(0.99n)!$ permutations
Suppose I have $n$ points $x_1,\dots,x_n$ that are all independent uniform samples in the unit square, and I'd like to find a short path (in terms of Euclidean length) that touches all of them (a ...
- 4,049
2
votes
2
answers
5k
views
Best fit for multiple shapes inside an area
Is there a forumla to come up with the best fit for multiple shapes inside a rectangular area, so that none of the shapes are overlapping?
- 121