The Mahler conjecture in two dimensions via the probabilistic method

TL;DR

This paper proves the 2D Mahler conjecture, which states that the Mahler volume is minimized by a cuboid, using a probabilistic approach involving random deletions in convex polygons.

Contribution

It provides a new probabilistic proof of the 2D Mahler conjecture, a longstanding open problem in convex geometry.

Findings

Deleting random edges or vertices reduces Mahler volume with positive probability

The proof confirms the conjecture in two dimensions

Introduces probabilistic methods to convex geometric problems

Abstract

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.