## 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 $d$?

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.

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### 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.