  In this note, our aim is to show that families of smooth hypersurfaces of
$\mathbb R^{n+1}$ which are all $C^1$--close enough to a fixed compact,
embedded one, have uniformly bounded constants in some relevant inequalities
for mathematical analysis, like Sobolev, Gagliardo-Nirenberg and ``geometric''
Calder\'on-Zygmund inequalities. This technical result is quite useful, in
particular, in the study of the geometric flows of hypersurfaces.
