Best approximation of functions by log-polynomials

Lasserre [La] proved that for every compact set K _ Rn and every even number d there exists a unique homogeneous polynomial g0 of degree d with K _ G1(g0) = fx 2 Rn : g0(x) _ 1g minimizing jG1(g)j among all such polynomials g fulfilling the condition K _ G1(g). This result extends the notion of the Löwner 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’