  We extend the notion of the smallest volume ellipsoid containing a convex
body in~$\mathbb{R}^{d}$ to the setting of logarithmically concave functions.
We consider a vast class of logarithmically concave functions whose superlevel
sets are concentric ellipsoids. For a fixed function from this class, we
consider the set of all its "affine" positions. For any log-concave function
$f$ on $\mathbb{R}^{d},$ we consider functions belonging to this set of
"affine" positions, and find the one with the smallest integral under the
condition that it is pointwise greater than or equal to $f.$ We study the
properties of existence and uniqueness of the solution to this problem. For any
$s \in [0,\infty),$ we consider the construction dual to the recently defined
John $s$-function \cite{ivanov2020functional}. We prove that such a
construction determines a unique function and call it the \emph{L\"owner
$s$-function} of $f.$ We study the L\"owner $s$-functions as $s$ tends to zero
and to infinity. Finally, extending the notion of the outer volume ratio, we
define the outer integral ratio of a log-concave function and give an
asymptotically tight bound on it. \end{abstract}

Functional Analysis