Help the (skinny) Hedgehogs | 31 Oct. 2019 |

In this first post we will see that the ‘‘almost surely’’ in the title is not only hinting at stochasticity.

For a long time this picture - as taken from Gabriel Peyré his outstanding twitter feed - has been on a wall in our local wildlife (inc. hedgehogs) shelter. This has lead to a few fun discussions and here we try to explain the picture a bit more.

Let B^{ell_1^n}_1 := {xin mathbf{R}^n;:;sum^n_{i=1}|x_i|leq 1}. Now, we want to find the largest ball B^{ell_2^n}_r:={xin mathbf{R}^n;:; sum^n_{i=1}x_i^2leq r^2} within B^{ell_1^n}_1. We can find that min_{xin partial B^{ell_1^n}_1}|x|_2 is given by |x_i^{star}|=n^{-1};forall iin overline{n} with |x^{star}|_2=sqrt{1/n}. This follows the most easily from geometric intuition.

Hyperplane P 

To prove it, see that B^{ell_1^n}_1 has 2^n ‘‘similar’’ faces, without loss of generality we take the one in the positive orthant of mathbf{R}^n. Now we fit a n-1-dimensional plane P (hyperplane) to this face and find the point pin P the closest (in ell_2-norm) to 0. This hyperplane can be parametrized by P={xin mathbf{R}^n;:;a^{top}x=b} for a=(1,dots,1)in mathbf{R}^n and b=1. Thus the normal (direction) is given by a. To find p, observe that for c=n^{-1} we have ca=:pin P. Hence we have B^{ell_2^n}_{sqrt{1/n}}subset B^{ell_1^n}_1. Now, let e_1:=(1,0dots,0)in mathbf{R}^n and define the other n-1 unit vectors correspondingly. Then, since e_iin B^{ell_1^n}_1 for any n we can think of a (skinny) hedgehog indeed for nto infty. The inscribed ball shrinks to 0, while the spikes remain.

Besides being a fun visualization of concentration, this picture means a bit more to me. For years I have been working at the Wildlife shelter in Delft and the moment has come where the lack of financial support from surrounding municipalities is becoming critical. As usual, this problem can be largely attributed to bureaucracy and the exorbitant amount of managers this world has. Nevertheless, the team in Delft and their colleagues around the country do a fantastic job in spreading the word (see Volkskrant, RTLnieuws). Lets hope Delft, and it neighbours, finally see value in protecting the local wildlife we still have.