  We study various generalizations of concentration of measure on the unit
sphere, in particular by means of log-Sobolev inequalities. First, we show
Sudakov-type concentration results and local semicircular laws for weighted
random matrices. A further branch addresses higher order concentration (i.\,e.,
concentration for non-Lipschitz functions which have bounded derivatives of
higher order) for $\ell_p^n$-spheres. This is based on a type of generalized
log-Sobolev inqualities referred to as $\mathrm{LS}_q$-inequalities. More
generally, we prove higher order concentration bounds for probability measures
on $\mathbb{R}^n$ which satisfy an $\mathrm{LS}_q$-inequality. Finally, we
derive concentration bounds for sequences of smooth symmetric functions on the
Euclidean sphere which are closely related to Edgeworth-type expansions.
