  Let $(R, \mathfrak{m})$ be a Noetherian local ring. This paper concerns
several extremal invariants arising from the study of the relation between
colength and (Hilbert--Samuel or Hilbert--Kunz) multiplicity of an
$\mathfrak{m}$-primary ideal. We introduce versions of these invariants by
restricting to various closures and ``cross-pollinate'' the two multiplicity
theories by asking for analogues invariants already established in one of the
theories.
  On the Hilbert--Samuel side, we prove that the analog of the
St\"{u}ckrad--Vogel invariant (that is, the infimum of the ratio between the
multiplicity and colength) for integrally closed $\mathfrak{m}$-primary ideals
is often $1$ under mild assumptions. We also compute the supremum and infimum
of the relative drops of multiplicity for (integrally closed)
$\mathfrak{m}$-primary ideals. On the Hilbert--Kunz side, we study several
analogs of the Lech--Mumford and St\"{u}ckrad--Vogel invariants.
