  We introduce the notion of Loewner (ellipsoid) function for a log concave
function and show that it is an extension of the Loewner ellipsoid for convex
bodies. We investigate its duality relation to the recently defined John
(ellipsoid) function by Alonso-Gutierrez, Merino, Jimenez and Villa. For convex
bodies, John and Loewner ellipsoids are dual to each other. Interestingly, this
need not be the case for the John function and the Loewner function.
