TY - JOUR

T1 - Mean Width of Regular Polytopes and Expected Maxima of Correlated Gaussian Variables

AU - Kabluchko, Z.

AU - Litvak, A. E.

AU - Zaporozhets, D.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - An old conjecture states that among all simplices inscribed in the unit sphere, the regular one has the maximal mean width. We restate this conjecture probabilistically and prove its asymptotic version. 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. We establish several formulas conjectured by S. Finch on the projection length W of the regular cube, simplex, and crosspolytope onto a line with random direction. Finally, we prove distributional limit theorems for W as the dimension of the regular polytope goes to ∞. Bibliography: 25 titles.

AB - An old conjecture states that among all simplices inscribed in the unit sphere, the regular one has the maximal mean width. We restate this conjecture probabilistically and prove its asymptotic version. 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. We establish several formulas conjectured by S. Finch on the projection length W of the regular cube, simplex, and crosspolytope onto a line with random direction. Finally, we prove distributional limit theorems for W as the dimension of the regular polytope goes to ∞. Bibliography: 25 titles.

UR - http://www.scopus.com/inward/record.url?scp=85026823044&partnerID=8YFLogxK

U2 - 10.1007/s10958-017-3492-3

DO - 10.1007/s10958-017-3492-3

M3 - Article

AN - SCOPUS:85026823044

VL - 225

SP - 770

EP - 787

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 5

ER -