  Motivated by the central limit problem for convex bodies, we study normal
approximation of linear functionals of high-dimensional random vectors with
various types of symmetries. In particular, we obtain results for distributions
which are coordinatewise symmetric, uniform in a regular simplex, or
spherically symmetric. Our proofs are based on Stein's method of exchangeable
pairs; as far as we know, this approach has not previously been used in convex
geometry and we give a brief introduction to the classical method. The
spherically symmetric case is treated by a variation of Stein's method which is
adapted for continuous symmetries.
