Best approximation of functions by log-polynomials

Lasserre [32] proved that for every compact set K subset of R-n and every even number d there exists a unique homogeneous polynomial g0 of degree d with K subset of G(1)(g(0)) = {x is an element of R-n : g(0)(x) <= 1} minimizing |G(1)(g)| among all such polynomials g fulfilling the condition K subset of G(1)(g). This result extends the notion of the Lowner 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'. (c) 2021 The Author(s). Published by Elsevier Inc.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).