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

Combinatorics and more 2018-03-12

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.