Dual $L_p$ John ellipsoids for general measures

Abstract

The dual $L_p$ John ellipsoids form a family that includes the origin-centered Löwner and Legendre ellipsoids. In this paper, we extend this family to general measures within the dual weighted $L_p$ Brunn--Minkowski theory. For measures with locally integrable densities that are continuous in a neighborhood of the boundary of the given star body, we establish a variational formula for $L_p$-harmonic radial combinations. The associated boundary integral leads to an optimization problem that remains meaningful under weaker assumptions on the boundary trace of the density. We prove the existence, uniqueness, and characterization of its solution. The resulting ellipsoids include the dual $L_p$ John ellipsoids for Gaussian measures. We also establish a monotonicity inequality, a measure inequality for homogeneous densities with its equality conditions on the weighted support, and an $L_p$ Löwner inclusion.

arXiv comment21 pages