Biggest Little Polyhedron

geometry open-problem combinatorial-geometry extremal-combinatorics polyhedra

Solution 1:

NEW -- I've posted Biggest Little Polyhedron—New Solutions in Combinatorial Geometry, which has a write up of many solutions.

Mathematica 10 allows Volume[ConvexHullMesh[points]], so I took a stab at the problem with annealling at biggest little polyhedron. In the image below, the red lines indicate a distance of 1.

None of these solutions are exact or perfected, and annealling is vulnerable to local maxima, so I expect improvements on all of these.

enter image description here

Related

Atiyah-MacDonald help with exercise 5.10
The origin/use of "derivative" and "differentiate"
Fully factored integer polynomials with constant differences
Evaluating a double integral by using $\int\arctan (1+\sin 2t)\ dt/(1+\sin 2t)$
Help me organize these concepts -- KKT conditions and dual problem
Why do you need to introduce complex numbers in order to solve cubics by radicals?
Why can I make a non-injective variable substitution?
there exists a sequence $x_n$ such that $\| x_n \|=1$ and $x_n$ converges weakly to $0$.
Is the multiplicative structure of a totally ordered field unique?
How can you describe topology to a non-mathematician without using continuous deformations?
$1/2$ or $1$? probability that all bacteria will die
Closed form of $\sum \frac{x^n}{n^n}$