Mean width of regular polytopes and expected maxima of correlated Gaussian variables
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 23, Tome 442 (2015), pp. 75-96
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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 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 $\infty$.
@article{ZNSL_2015_442_a3,
author = {Z. Kabluchko and A. E. Litvak and D. Zaporozhets},
title = {Mean width of regular polytopes and expected maxima of correlated {Gaussian} variables},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {75--96},
publisher = {mathdoc},
volume = {442},
year = {2015},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_442_a3/}
}
TY - JOUR AU - Z. Kabluchko AU - A. E. Litvak AU - D. Zaporozhets TI - Mean width of regular polytopes and expected maxima of correlated Gaussian variables JO - Zapiski Nauchnykh Seminarov POMI PY - 2015 SP - 75 EP - 96 VL - 442 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/ZNSL_2015_442_a3/ LA - en ID - ZNSL_2015_442_a3 ER -
%0 Journal Article %A Z. Kabluchko %A A. E. Litvak %A D. Zaporozhets %T Mean width of regular polytopes and expected maxima of correlated Gaussian variables %J Zapiski Nauchnykh Seminarov POMI %D 2015 %P 75-96 %V 442 %I mathdoc %U http://geodesic.mathdoc.fr/item/ZNSL_2015_442_a3/ %G en %F ZNSL_2015_442_a3
Z. Kabluchko; A. E. Litvak; D. Zaporozhets. Mean width of regular polytopes and expected maxima of correlated Gaussian variables. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 23, Tome 442 (2015), pp. 75-96. http://geodesic.mathdoc.fr/item/ZNSL_2015_442_a3/