An Application of Shadow Systems to Mahler's Conjecture

We elaborate on the use of shadow systems to prove a particular case of the conjectured lower bound of the volume product P(K) = minz∈int(K) |K|||K|, where K ⊂ R is a convex body and K = {y ∈ R : (y − z) · (x − z) 6 1 for all x ∈ K} is the polar body of K with respect to the center of polarity z. In particular, we show that if K ⊂ R is the convex hull of two 2-dimensional convex bodies, then P(K) > P(∆), where ∆ is a 3-dimensional simplex, thus confirming the 3-dimensional case of Mahler conjecture, for this class of bodies. A similar result is provided for the symmetric case, where we prove that if K ⊂ R is symmetric and the convex hull of two 2-dimensional convex bodies, then P(K) > P(B ∞), where B ∞ is the unit cube.