  The "Mahler volume" is, intuitively speaking, a measure of how "round" a
centrally symmetric convex body is. In one direction this intuition is given
weight by a result of Santalo, who in the 1940s showed that the Mahler volume
is maximized, in a given dimension, by the unit sphere and its linear images,
and only these. A counterpart to this result in the opposite direction is
proposed by a conjecture, formulated by Kurt Mahler in the 1930s and still open
in dimensions 4 and greater, asserting that the Mahler volume should be
minimized by a cuboid. In this article we present a seemingly new proof of the
2-dimensional case of this conjecture via the probabilistic method. The central
idea is to show that either deleting a random pair of edges from a centrally
symmetric convex polygon, or deleting a random pair of vertices, reduces the
Mahler volume with positive probability.
