  Consider a set represented by an inequality. An interesting phenomenon which
occurs in various settings in mathematics is that the interior of this set is
the subset where strict inequality holds, the boundary is the subset where
equality holds, and the closure of the set is the closure of its interior. This
paper discusses this phenomenon assuming the set is a Voronoi cell induced by
given sites (subsets), a geometric object which appears in many fields of
science and technology and has diverse applications. Simple counterexamples
show that the discussed phenomenon does not hold in general, but it is
established in a wide class of cases. More precisely, the setting is a
(possibly infinite dimensional) uniformly convex normed space with arbitrary
positively separated sites. An important ingredient in the proof is a strong
version of the triangle inequality due to Clarkson (1936), an interesting
inequality which has been almost totally forgotten.
