This article introduces
versions of the support function of a convex body
and associates to these
canonical
-polar bodies
and Mahler volumes
. Classical polarity is
then seen as
-polarity.
This one-parameter generalization of polarity leads to a generalization of the Mahler conjectures,
with a subtle advantage over the original conjecture: conjectural uniqueness of extremizers
for each
.
We settle the upper bound by demonstrating the existence and uniqueness of an
-Santaló point
and an
-Santaló
inequality for symmetric convex bodies. The proof uses Ball’s
Brunn–Minkowski inequality for harmonic means, the classical
Brunn–Minkowski inequality, symmetrization, and a systematic study of the
functionals. Using our
results on the
-Santaló
point and a new observation motivated by complex geometry, we show how
Bourgain’s slicing conjecture can be reduced to lower bounds on the
-Mahler
volume coupled with a certain conjectural convexity property
of the logarithm of the Monge–Ampère measure of the
-support
function. We derive a suboptimal version of this convexity using Kobayashi’s
theorem on the Ricci curvature of Bergman metrics to illustrate this approach
to slicing. Finally, we explain how Nazarov’s complex-analytic approach
to the classical Mahler conjecture is instead precisely an approach to the
-Mahler
conjecture.
Keywords
support function, isotropic constant, Bergman kernel, Ricci
curvature, Mahler conjecture, hyperplane conjecture,
slicing problem
