  A new position is introduced and studied for the convolution of log-concave
functions, which may be regarded as a functional analogue of the maximum
intersection position of convex bodies introduced and studied by
Artstein-Avidan and Katzin (2018) and Artstein-Avidan and Putterman (2022). Our
main result is a John-type theorem for the maximal intersection position of a
pair of log-concave functions, including the corresponding decomposition of the
identity. The main result holds under very weak assumptions on the functions;
in particular, the functions considered may both have unbounded supports.
