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

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