Mean Width of Regular Polytopes and Expected Maxima of Correlated Gaussian Variables
نویسندگان
چکیده
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 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 ≤ E max{|η1|, . . . , |ηn|} E max{η1, . . . , η2n} ≤ min {√ 2n 2n− 1 , 1 + C n logn } , 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 ∞. In the case of the ndimensional unit cube Qn, we prove that WQn − √ 2n π d −→ n→∞ N ( 0, π − 3 π ) , whereas for the simplex and the crosspolytope the limiting distributions are related to the Gumbel double exponential law. 1. Conjecture on the mean width 1.
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