  We prove the four-dimensional Gaussian random vector maximum conjecture. This
conjecture asserts that among all centered Gaussian random vectors
$X=(X_1,X_2,X_3,X_4)$ with $E[X_i^2]=1$, $1\le i\le 4$, the expectation
$E[\max(X_1,X_2,X_3,X_4)]$ is maximal if and only if all off-diagonal elements
of the covariance matrix equal $-\frac{1}{3}$. As a direct consequence, we
resolve the three-dimensional simplex mean width conjecture. This latter
conjecture is a long-standing open problem in convex geometry, which asserts
that among all simplices inscribed into the three-dimensional unit Euclidean
ball the regular simplex has the maximal mean width.
