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We prove the four-dimensional Gaussian random vector maximum conjecture. Thisconjecture 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 elementsof the covariance matrix equal $-\frac{1}{3}$. As a direct consequence, weresolve the three-dimensional simplex mean width conjecture. This latterconjecture is a long-standing open problem in convex geometry, which assertsthat among all simplices inscribed into the three-dimensional unit Euclideanball the regular simplex has the maximal mean width.

The 4-D Gaussian Random Vector Maximum Conjecture and the 3-D Simplex Mean Width Conjecture