  The Brydges-Imbrie dimensional reduction formula relates the pressure of a
$d$-dimensional gas of hard spheres to a model of $(d+2)$-dimensional branched
polymers. Brydges and Imbrie's proof was non-constructive and relied on a
supersymmetric localization lemma. The main result of this article is a
constructive proof of a more general dimensional reduction formula that
contains the Brydges--Imbrie formula as a special case. Central to the proof
are invariance lemmas, which were first introduced by Kenyon and Winkler for
branched polymers. The new dimensional reduction formulas rely on invariance
lemmas for central hyperplane arrangements that are due to M\'esz\'aros and
Postnikov.
  Several applications are presented, notably dimensional reduction formulas
for (i) non-spherical bodies and (ii) for corrections to the pressure due to
symmetry effects.
