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

Accumulated Random Distances in High Dimensions -- Ways of Calculation