Two short proofs of incompatibility for correlation matrices

Source abstract

Let $P_d$ be the elliptope of $d \times d$ correlation matrices, and, for a standardised law $F$, let $S_d^F$ be the set of correlation matrices of random vectors with all margins $F$. We give two short proofs. First, an eleven-term representation of $|x|^4$ as a sum of fourth powers of linear forms on $\mathbb{R}^4$ yields $S_{11} \neq P_{11}$ for uniform margins. Second, the six diagonals of the regular icosahedron yield $S_6^F \neq P_6$ for arcsine margins. Together with a result of Devroye and Letac, this gives $S_d^F = P_d$ if and only if $d \le 5$, settling a conjecture of theirs.