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 mode 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: Doest the claim in ``Question 2'' hold for arbitrary probability distributions?

Any help or insight about these questions would be appreciated!

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!