Best approximation of functions by log-polynomials
Abstract
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)=\x∈ Rn:g0(x)≤ 1\ minimizing |G1(g)| among all such polynomials g fulfilling the condition K⊂ G1(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'.