Turns out that I am still on a recreational mathematics run. Here is one I have been working on, arising from trying to explain norms and data science.
Barry Rowlingson and John Mount asked the following question.
v2in Rn with each coordinate generated IID normal mean zero, standard deviation 1. This is a common way to generate vectors with a uniform spherical distribution. Let
pndenote the probability that
(||v1||1 ≥ ||v2||1) = (||v1||2 ≥ ||v2||2). What is
limn → ∞ pn?
It turns out the answer is:
1/2 + arctan(1/sqrt(π - 3)) / π ≅ 0.8854404657887897. I’ve taken to calling this the “L1L2 AUC” or concordance. This is not the first value I guessed.
The rather long (and brutal) argument chain to establish this can be found here. Along the way we had to solve for the expected L1 norm of a vector with unit L2 norm, and also work out
P[(X + Y ≥ 0) = (X ≥ 0)] (for X, Y independent mean zero normal random variables with known variances, we call this the sign tilting lemma).
It was great to get the old “conjecture and prove/disprove” engine spinning again.