On the upper bound for the mathematical expectation of the norm of a vector uniformly distributed on the sphere and the phenomenon of concentration of uniform measure on the sphere

TL;DR

This paper derives upper bounds for the expected q-norm of vectors uniformly distributed on the sphere, highlighting measure concentration phenomena, supported by numerical experiments.

Contribution

It provides new upper bounds for the expected q-norm of spherical vectors and explores measure concentration effects.

Findings

Derived upper bounds for q-norm expectations

Numerical experiments support theoretical results

Illustrated measure concentration on the sphere

Abstract

We considered the problem of obtaining upper bounds for the mathematical expectation of the $q$-norm ($2\leqslant q \leqslant \infty$) of the vector which is uniformly distributed on the unit Euclidean sphere. We finish the paper with numerical experiments illustrating our results.