  Small ball inequalities have been extensively studied in the setting of
Gaussian processes and associated Banach or Hilbert spaces. In this paper, we
focus on studying small ball probabilities for sums or differences of
independent, identically distributed random elements taking values in very
general sets. Depending on the setting--abelian or nonabelian groups, or vector
spaces, or Banach spaces--we provide a collection of inequalities relating
different small ball probabilities that are sharp in many cases of interest. We
prove these distribution-free probabilistic inequalities by showing that
underlying them are inequalities of extremal combinatorial nature, related
among other things to classical packing problems such as the kissing number
problem. Applications are given to moment inequalities.
