Around the simplex mean width conjecture

In this note we discuss an old conjecture in Convex Geometry asserting that the regular simplex has the largest mean width among all simplices inscribed into the Euclidean ball and its relation to Information Theory. Equivalently, in the language of Gaussian processes, the conjecture states that the expectation of the maximum of n + 1 standard Gaussian variables is maximal when the expectations of all pairwise products are −1/n, that is, when the Gaussian variables form a regular simplex in L2. We mention other conjectures as well, in particular on the expectation of the smallest (in absolute value) order statistic of a sequence of standard Gaussian variables (not necessarily independent), where we expect the same answer. AMS 2010 Classification: primary 52A23, 60G35, 94A12, 46B06; secondary 52A40, 60G15, 94A05, 46B09.