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An old conjecture states that among all simplices inscribed in the unitsphere the regular one has the maximal mean width. An equivalent formulation isthat for any centered Gaussian vector $(\xi_1,\dots,\xi_n)$ satisfying $\mathbbE\xi_1^2= \dots =\mathbb E\xi_n^2=1$ one has $$  \mathbb E\,\max\{\xi_1,\dots,\xi_n\}\leq\sqrt{\frac{n}{n-1}}\, \mathbbE\,\max\{\eta_1,\dots,\eta_n\}, $$ where $\eta_1,\eta_2,\dots,$ are independentstandard Gaussian variables. Using this probabilistic interpretation we derivean asymptotic version of the conjecture. We also show that the mean width ofthe regular simplex with $2n$ vertices is remarkably close to the mean width ofthe regular crosspolytope with the same number of vertices. Interpretedprobabilistically, our result states that $$  1\leq\frac{\mathbb E\,\max\{|\eta_1|,\dots,|\eta_n|\}}{\mathbbE\,\max\{\eta_1,\dots,\eta_{2n}\}}  \leq\min\left\{\sqrt{\frac{2n}{2n-1}}, \, 1+\frac{C}{n\, \log n} \right\}, $$where $C>0$ is an absolute constant. We also compute the higher moments of theprojection length $W$ of the regular cube, simplex and crosspolytope onto aline with random direction, thus proving several formulas conjectured by S.Finch. Finally, we prove distributional limit theorems for the length of randomprojection as the dimension goes to $\infty$. In the case of the$n$-dimensional unit cube $Q_n$, we prove that $$ W_{Q_n} -\sqrt{\frac{2n}{\pi}} \overset{d}{\underset{n\to\infty}\longrightarrow}{\mathcal{N}} \left(0, \frac{\pi-3}{\pi}\right), $$ whereas for the simplex andthe crosspolytope the limiting distributions are related to the Gumbel doubleexponential law.

Mean width of regular polytopes and expected maxima of correlated Gaussian variables