Orlicz-Legendre Ellipsoids

TL;DR

This paper introduces Orlicz-Legendre ellipsoids within the dual Orlicz Brunn-Minkowski theory, establishing new inequalities and linking their properties to measure isotropy, thus generalizing classical inertia ellipsoids.

Contribution

It is the first introduction of Orlicz-Legendre ellipsoids, expanding the dual Orlicz Brunn-Minkowski framework and connecting geometric ellipsoids with measure isotropy.

Findings

Established new affine isoperimetric inequalities.

Connected Orlicz-Legendre ellipsoids to measure isotropy.

Generalized classical Legendre ellipsoids of inertia.

Abstract

The Orlicz-Legendre ellipsoids, which are in the framework of emerging dual Orlicz Brunn-Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz-John ellipsoids, and have largely generalized the classical Legendre ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz-Legendre ellipsoids and isotropy of measures is demonstrated.