  Lasserre [La] proved that for every compact set $K\subset\mathbb R^n$ and
every even number $d$ there exists a unique homogeneous polynomial $g_0$ of
degree $d$ with $K\subset G_1(g_0)=\{x\in\mathbb R^n:g_0(x)\leq 1\}$ minimizing
$|G_1(g)|$ among all such polynomials $g$ fulfilling the condition $K\subset
G_1(g)$. This result extends the notion of the L\"owner ellipsoid, not only
from convex bodies to arbitrary compact sets (which was immediate if $d=2$ by
taking convex hulls), but also from ellipsoids to level sets of homogeneous
polynomial of an arbitrary even degree.
  In this paper we extend this result for the class of non-negative log-concave
functions in two different ways. One of them is the straightforward extension
of the known results, and the other one is a suitable extension with uniqueness
of the solution in the corresponding problem and a characterization in terms of
some 'contact points'.
