Best approximation of functions by log-polynomials

TL;DR

This paper extends Lasserre's geometric approximation results from convex sets to non-negative log-concave functions, providing new existence, uniqueness, and characterization results for polynomial level set approximations.

Contribution

It generalizes Lasserre's approximation framework from compact sets to log-concave functions, including uniqueness and contact point characterizations.

Findings

Extended approximation results to log-concave functions

Established uniqueness of polynomial level set approximations

Provided characterization via contact points

Abstract

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'.