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

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.