TITLE: Biggest Little Polyhedron
QUESTION [12 upvotes]: The Biggest Little Polygon problem asks for the polygon with greatest area where the largest diameter is 1.  Let's add a dimension and find the largest volumes.
What is the biggest little polyhedron that can fill space?
What is the biggest little hexahedron?

REPLY [6 votes]: 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.