  The goal of this paper is to present a lower bound for the Mahler volume of
at least 4-dimensional symmetric convex bodies. We define a computable
dimension dependent constant through a 2-dimensional variational (max-min)
procedure and demonstrate that the Mahler volume of every (at least
4-dimensional) symmetric convex body is greater than a (simple) function of
this constant. Similar to the proof of Gromov's Waist of the Sphere Theorem in
[18], our result is proved via localisation-type arguments obtained from a
suitable measurable partition (or partitions) of the canonical sphere.
