  We introduce a new way of representing logarithmically concave functions on
$\mathbb{R}^{d}$. It allows us to extend the notion of the largest volume
ellipsoid contained in a convex body to the setting of logarithmically concave
functions as follows. For every $s>0$, we define a class of non-negative
functions on $\mathbb{R}^{d}$ derived from ellipsoids in $\mathbb{R}^{d+1}$.
For any log-concave function $f$ on $\mathbb{R}^{d}$, and any fixed $s>0$, we
consider functions belonging to this class, and find the one with the largest
integral under the condition that it is pointwise less than or equal to $f$,
and we call it the \emph{\jsfunction} of $f$. After establishing existence and
uniqueness, we give a characterization of this function similar to the one
given by John in his fundamental theorem. We find that John $s$-functions
converge to characteristic functions of ellipsoids as $s$ tends to zero and to
Gaussian densities as $s$ tends to infinity.
  As an application, we prove a quantitative Helly type result: the integral of
the pointwise minimum of any family of log-concave functions is at least a
constant $c_d$ multiple of the integral of the pointwise minimum of a properly
chosen subfamily of size $3d+2$, where $c_d$ depends only on $d$.
