Thread by @gro_tsen

In the tweet below, I mention the variance of the median of three independent standard Gaussian variables to be 0.448…, and I say that it doesn't seem to have a closed form expression. But actually it does! It's 1 − (√3)/π. But now this opens a trove of new questions. •1/17

https://twitter.com/gro_tsen/status/1430202153164021763

Thanks to @ArthurB for pointing out this closed form expression to me (below). This reminded me I once knew the max of 4 iid standard Gaussian variables has expected value: (3/√π)·[½+asin(1/3)/π]. Where does this come from? •2/17

https://twitter.com/ArthurB/status/1430462416316518400

More generally, we might ask about the variance of the median of 2s+1 independent standard Gaussian variables (for s∈ℕ). Or we might ask about the expected value of the maximum of r such variables (for r∈ℕ). Can we find closed form formulæ for these? •3/17

Well, I don't have a satisfactory answer (I mean not even like “it's a well-known open problem”), and somewhat amazingly, nobody seems to have one, even though the problem relates to other famous ones — but the literature is all over the place. Here's what I DO understand. •4/17

Here's the thing: suppose X := (X_1,…,X_n) are n independent standard Gaussian variables. The questions I mention above essentially reduce to computing the integral of some monomial m(X) in the X_i (times the Gaussian distribution) over a linear polyhedral cone C: … •5/17

… meaning a cone defined by linear inequalities on the X_i. For example if h(X) is the Gaussian distribution (exp(−|X|²/2)/(2π)^(n/2)), the expected value of the max of X_1,…,X_n is n times the integral of X₁·h(X) over the cone defined by X_1 ≥ X_i for all i … •6/17

… (because this cone C defines the event “X_1 is the largest”, whose probability is ∫_C h(X) = 1/n by symmetry, and we want the expected value n·∫_C X₁·h(X) of X_1 under this event). So all reduces to computing integrals of the form ∫_C m(X)·h(X). How? •7/17

Well, ignore the monomial m(X) for the time being, and just concentrate on the special case of the (“powerless”) integral ∫_C h(X). This amounts to asking what's the probability that the X_i satisfy certain linear inequalities (defining C): can we compute this? •8/17

It turns out that this is a fairly famous problem (Schläfli): the thing is, the integral separates into radial part, which doesn't involve C so it's trivial, and angular part. So we're just trying to compute the volume of C intersected by the (n−1)-dimensional sphere. •9/17

In other words, the problem is: how do we compute the volume of a simplex (the locus bounded by n hyperplanes in general position) in an (n−1)-sphere? There is a completely analogous problem of computing the volume of a simplex in hyperbolic geometry. •10/17

Many people probably know that the area of a spherical triangle (on the 2-sphere) is given by the sum of its angles minus π. (There is an analogous formula for the hyperbolic triangle.) But what about a simplex (tetrahedron) in the 3-sphere? Answer involves dilogarithms. •11/17

For such a formula in 3D, see: Murakami & Yano, “On the Volume of a Hyperbolic and Spherical Tetrahedron” (Comm. Anal. Geom. 13 (2005), 379–400) intlpress.com/site/pub/files…‌; see also Murakami, “The volume formulas for a spherical tetrahedron” arxiv.org/abs/1011.2584 •12/17

The general problem of computing the volume of a spherical simplex dates back to Schläfli and Aomoto. See Kohno, “The Volume of a Hyperbolic Simplex and Iterated Integrals” doi.org/10.1142/978981…‌, available at ms.u-tokyo.ac.jp/~kohno/papers/…‌, for a survey and references. •13/17

But I still don't understand what is known about closed forms. The state of the art may be Kellerhals' “Volumes in Hyperbolic 5-space”, Geom. Funct. Anal. 5 (1995), 640–667, link.springer.com/article/10.100…‌, available at researchgate.net/publication/22…‌, using trilogarithms. •14/17

Now remember that here I'm just talking about the (“powerless”) integral ∫_C h(X), the volume of a simplex, but the general problem is to compute a (“powerful”) integral of the form ∫_C m(X)·h(X) involving some monomial m(X) in the X_i, so higher moments of the simplex. •15/17

Well, it turns out that the latter problem can be reduced to the former. Amazingly, the only reference available which discusses it in any detail is this web page by Warren D. Smith which is about sth quite different (“range voting mechanisms”): rangevoting.org/BestVrange.htm… •16/17

So, to summarize, a question such as “what is the expected maximum of r iid standard Gaussian variables?” can be reduced, by techniques found in an unpublished online text, to the Schläfli problem of computing simplex volumes in closed form, whose status is unclear. 😕 •17/17