  In this note we refine and improve some of the calculations in our 2019
article with Yair Censor (Applied Mathematics and Optimization, accepted for
publication) where an analysis of the superiorization method is made via the
principle of concentration of measure. Some paragraphs there are repeated here
for the sake of completeness. Yet, for the case of accumulating 'steps' on the
sphere, reference to distances as done there is replaced by reference to the
angles, which makes simpler expressions. The treatment here of the action of a
random transformation is also rather 'cleaner'. For some standard deviations,
precise inequalities are here obtained rather then just O-expressions. Some
further settings are mentioned, of no direct interest as per the latter
article, showing results that similar calculations yield.
