An old conjecture states that among all simplices inscribed in the unit sphere the regular one has the maximal mean width. An equivalent formulation is that for any centered Gaussian vector $(ξ_1,\dots,ξ_n)$ satisfying $\mathbb Eξ_1^2= \dots =\mathbb Eξ_n^2=1$ one has $$ \mathbb E\,\max\{ξ_1,\dots,ξ_n\}\leq\sqrt{\frac{n}{n-1}}\, \mathbb E\,\max\{η_1,\dots,η_n\}, $$ where $η_1,η_2,\dots,$ are independent standard Gaussian variables. Using this probabilistic interpretation we derive an asymptotic version of the conjecture. We also show that the mean width of the regular simplex with $2n$ vertices is remarkably close to the mean width of the regular crosspolytope with the same number of vertices. Interpreted probabilistically, our result states that $$ 1\leq\frac{\mathbb E\,\max\{|η_1|,\dots,|η_n|\}}{\mathbb E\,\max\{η_1,\dots,η_{2n}\}} \leq\min\left\{\sqrt{\frac{2n}{2n-1}}, \, 1+\frac{C}{n\, \log n} \right\}, $$ where $C>0$ is an absolute constant. We also compute the higher moments of the projection length $W$ of the regular cube, simplex and crosspolytope onto a line with random direction, thus proving several formulas conjectured by S. Finch. Finally, we prove distributional limit theorems for the length of random projection as the dimension goes to $\infty$. In the case of the $n$-dimensional unit cube $Q_n$, we prove that $$ W_{Q_n} - \sqrt{\frac{2n}π} \overset{d}{\underset{n\to\infty}\longrightarrow} {\mathcal{N}} \left(0, \frac{π-3}π\right), $$ whereas for the simplex and the crosspolytope the limiting distributions are related to the Gumbel double exponential law.