Functional John and Löwner Conditions for Pairs of Log-Concave Functions

Abstract John’s fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ $\mathbb{R}^{d}$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids positions (affine images) of another $L$. Another, more recent direction consider logarithmically concave functions on instead bodies: we designate some special, radially symmetric log-concave function $g$ as analogue Euclidean ball, want find its integral position under constraint that it pointwise below given $f$. We follow both directions simultaneously: functional question, allow essentially any meaningful play role above. Our general theorems jointly extend known results directions. The dual problem setting bodies asks for smallest ellipsoid, called Löwner’s containing $K$. analogous functions: characterize solutions optimization finding above It turns out setting, relationship between John Löwner problems intricate than bodies.