An origin-symmetric convex body $K$ in $\mathbb{R}^n$ is said to be in the John position if the maximal volume ellipsoid contained in it is the Euclidean ball. One of the most celebrated theorems in geometric functional analysis is John's theorem, which says that if a convex body $K$ is in John position, then there are a collection of points $u_1,\dots u_m \in S^{n-1} \cap \partial K$ and positive scalars $c_1,\dots, c_m$ for which $I_n = \sum_{j=1}^m c_j u_j \otimes u_j$, where $I_n$ is the identity operator in $\mathbb{R}^n$.  This theorem has numerous consequences such as an estimate for the Banach-Mazur distance to Euclidean space  and the reverse isoperimetric inequality due to Ball. 

Recently, much attention has been given to translating notions from convex geometry and geometric functional analysis to the world of log-concave functions.  In particular, over the last 10 years, extensions of these celebrated theorems (in some forms) have been translated to the world of log-concave functions.

In this talk, we focus on a complementary set of positions, called "maximal-intersection positions," originally introduced by Artstein-Avidan and Katzin in 2016. This collection of position includes the John (and its dual the Lowner position) as particular cases.  The extension of this notion to the world of functions is seen through the following extremal problem for the convolution of a pair of functions: given a pair of integrable log-concave functions $f$ and $g$, find

$\sup_{(T,b) \in SL_n(\mathbb{R}) \times \mathbb{R^n}} \int f(x) g(T^{-1}x-T^{-1}b)dx.$

This is based on a joint work with Steven Hoehner.