Warut wrote:

<< Ian Stewart's NewScientist article on the sausage conjecture:

http://www.newscientist.com/article/mg20026873.800-the-scientific-guide-to-g...

...

...

Thanks for posting this, Warut -- it's totally fascinating!

(I recall hearing about the situation through 4 dimensions from Neil Sloane some years back.)

What is most fascinating about this is that in dimensions up through 4 (well, OK, just 2, 3, and 4), the least-volume convex set containing n unit balls is a sausage up through some value of n, and then it becomes a more ball-shaped mass.

But in sufficiently high dimensions (currently known for dim >= 42), the sausage (straight-line) configuration of balls is optimals no matter how many balls are involved.

Maybe it's just me, but I was astonished to learn even that for 4 unit balls in 3-space, the tetrahedral configuration is not the one whose convex hull has the least volume; it's all 4 balls in a straight line.

--Dan

What I find hardest to believe is the claim that there's only one cutover at 57 spheres in the 3D case. I'd expect it to jitter for a while for 58, 59, ... . --rwg ALGORISMIC MICROGLIAS