Best approximation of functions by log-polynomials.

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 Lowner ellipsoid, not only from centrally symmetric convex bodies to any compact set, but also from ellipsoids to level sets of homogeneous polynomial of an arbitrary even degree. In this paper we extend this result to the class of non-negative integrable log-concave functions in two different ways, by considering two different minimization problems. In one case we obtain a straightforward extension of the known results. In the other case we obtain a suitable extension of the Lowner ellipsoid with uniqueness of solution in the corresponding minimization problem and a characterization of this solution in terms of some `contact points'.