  In this short paper the discussion of the pointwise characterization of
functions $f$ in the Sobolev space $W^{m,p}(\R^n)$ given in the recent paper
(Bojarski) is supplemented in \SS1 by a direct, essentially geometric, proof of
the novel inequality (for $m>1$), appearing in Bojarski apparently for the
first time, and involving the use of the $m$-th difference of the function $f$.
  Moreover in \SS2 some additional comments to the text in Bojarski are given
and a natural class of Sobolev spaces in domains $G$ in $\R^n$ is defined. \SS3
contains some final remarks.
