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Holes in Random Point Sets

This website provides programs to verify the computer-assisted results from "Tight bounds on the expected number of holes in random point sets".

Short description of the programs polyhedron.sage and ball.sage

The programs "polyhedron.sage" and "ball.sage" can be used to obtain estimates on the constants $$c_{3,4}^K=\lim_{n \to \infty} n^{-3}EH_{3,4}^K(n)$$ when $K$ is a Platonic solid or a ball. The idea is to repeatedly sample 3 points $p_1,p_2,p_3$ uniformly at random from $K$ and to compute the ratio of the so-called "Fischer triangle" spanned by $p_1,p_2,p_3$ which is contained inside $K$. This ratio coincides with $c_{3,4}^K$; see Section 3 in our previous paper. For the ball, run 
sage ball.sage
For the tetrahedron, cube, octahedron, dodecahedron, or icosahedron, run 
sage polyhedron.sage [t/c/o/d/i]

Short description of the program test_planar_holes.py

The program samples $t$ sets of $n$ random points from a triangle, a square, or a disk and computes the average number of $k$-gons. More specifically, the points are sampled uniformly among the $[-g,+g] \times [-g,+g]$ grid. If the shape is "triangle", $x$-coordinate should be less than the $y$-coordinate, and if the shape is "disk", the distance to the origin is at most $g$. Sampled points which do not fulfill these conditions are omited. To run the program use 
python test_planar_holes.py g k n t [ball/triangle/square]

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Last update: 20.11.2021 by Manfred Scheucher