2020/07/15 by Alonso-Gutiérrez, David, Merino, Bernardo González, Villa, Rafael · 1 citation
#46B20 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 52A21 #Secondary 52A40
paper · doi:10.48550/arxiv.2007.07952
Lasserre [La] proved that for every compact set K⊂\mathbb Rn and every even number d there exists a unique homogeneous polynomial g0 of degree d with K⊂ G1(g0)=\x∈\mathbb 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 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'.