In the process of computing inclusion constants for the complex matrix cube (which is a free spectrahedron), the following identity was proven: for all $n \geq 1$,
$$\mathbb E \Big| \sum_{i=1}^{2n} x_i^2 - \sum_{j=1}^{2n} y_j^2 \Big| = \mathbb E \Big| \sum_{i=1}^{2n} x_i^2 - \sum_{j=1}^{2n-2} y_j^2 \Big| = 4^{1-n}n \binom{2n}{n},$$
where $x_i$,$y_j$ are i.i.d. standard real Gaussian random variables. The proof we have at this moment is by using the explicit form of the density of the difference of two $\chi^2$ random variables, see here and also *Klar, Bernhard*, **A note on gamma difference distributions**, ZBL07183251.

Since the result is so simple, there should be a more direct and more insightful proof of it.

**Question 1:** give an easy, conceptual proof of the identity above.

Consider now the function $$k \mapsto \mathbb E \Big| \sum_{i=1}^{2k} x_i^2 - \sum_{j=1}^{2(n-k)} y_j^2 \Big|$$ from $\{0,1,\ldots, n\} \to \mathbb R_+$; as above, the $x_i$ and $y_j$ are standard i.i.d. Gaussians.

**Question 2:** Show that the function above is unimodular, and that its minimum is attained at $k = \lfloor n/2 \rfloor$.

There is a pretty involved proof of the second fact above in *Helton, J. William; Klep, Igor; McCullough, Scott; Schweighofer, Markus*, **Dilations, linear matrix inequalities, the matrix cube problem and beta distributions**, Memoirs of the American Mathematical Society 1232. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3455-7/pbk; 978-1-4704-4947-6/ebook). vi, 106 p. (2019). ZBL1447.47009.

**Question 3:** Does the claim in ``Question 2'' hold for arbitrary probability distributions?

Any help or insight about these questions would be appreciated!