Mean Width of Regular Polytopes and Expected Maxima of Correlated Gaussian Variables

Mean Width of Regular Polytopes and Expected Maxima of Correlated Gaussian Variables

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 (ξ1, . . . , ξn) satisfying Eξ2 1 = · · · = Eξ2 n = 1 one has E max{ξ1, . . . , ξn} ≤ √ n n− 1 E max{η1, . . . , ηn}, where η1, η2, . . . , are independent standard Gaussian variables. Using this probabilist...

Expected Number of Local Maxima of Some Gaussian Random Polnomials

expected number of local maxima of some gaussian random polnomials

Asymptotic mean values of Gaussian polytopes

We consider geometric functionals of the convex hull of normally distributed random points in Euclidean space R. In particular, we determine the asymptotic behaviour of the expected value of such functionals and of related geometric probabilities, as the number of points increases.

Expected Number of Local Maxima of Some Gaussian Random Polynomials

Let Qn(x) = ∑n i=0 Aix i be a random algebraic polynomial where the coefficients A0, A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments ∆j = Aj − Aj−1, j = 0, 1, 2, · · · are independent, A−1 = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. We study the asymptotic behaviour of the expected number of lo...