Test your intuition 33: Why is the density of any packing of unit balls decay exponentially with the dimension?

Test your intuition: What is the simplest explanation you can give to the fact that the density of every packing of unit balls in R^d is exponentially small in d?

Answers are most welcome.

Of course, understanding the asymptotic behavior of the density \rho_d of densest packing of unit spheres in R^d is a central  problem in geometry. It is a long standing hope (perhaps naïve) that algebraic-geometry codes will eventually lead to examples showing that \rho_d \ge (2-\delta)^{-d} giving an exponential improvement of Minkowsky’s 1905 bound.  (For more on sphere packing asymptotically and in dimensions 8 and 24 see this post.)

The result by Serge Vlăduţ from the previous post can be seen (optimistically) as a step in the direction of exponential improvement to Minkowski’s bound.

 

 

Advertisements
This entry was posted in Combinatorics, Geometry, Test your intuition and tagged , . Bookmark the permalink.

4 Responses to Test your intuition 33: Why is the density of any packing of unit balls decay exponentially with the dimension?

  1. Craig says:

    Its density is exponential small because its volume is exponentially small.

    • shacharlovett says:

      but then the same can be said about tiling R^n with cubes [0,1/2]^n, whose volumes also exponentially decrease

      • Craig says:

        Yes, but when tiling with cubes [-1/2,1/2]^n, the volume stays the same, but the volume of the ball with diameter 1 in [-1/2,1/2]^n is exponentially small.

  2. John Sullivan says:

    I’d say it is because the regular n-simplex is very spiky for large n, so most of its volume is far from the corners. In particular if we place unit balls at the vertices of a simplex of side length 2, then they cover only an exponentially small fraction of the volume of the simplex. And I think it was Rogers who pointed out that this ratio is an upper bound for the density of any packing.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

w

Connecting to %s