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arXiv · 2608.21958

Dual $L_p$ John ellipsoids for general measures

Ai-Jun Li·Baihe Liu·Qingzhong Huang
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Abstract

The dual $L_p$ John ellipsoid, including the classical L\"{o}wner and Legendre ellipsoids, arises as the solution to a certain optimization problem. In this paper, we consider a broad extension within the framework of the dual weighted $L_p$ Brunn-Minkowski theory. A variational formula for general measures with locally integrable densities, under $L_p$-harmonic radial combinations, is established. This leads us to propose a corresponding optimization problem for general measures. We prove the existence, uniqueness and characterization of the solution to this problem, which defines a new type of ellipsoid. This ellipsoid extends the dual $L_p$ John ellipsoid to a significantly general more setting, including Gaussian measures. The related geometric inequalities and the $L_p$ L\"{o}wner inclusion for general measures are also given.

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Ai-Jun Li, Baihe Liu, Qingzhong Huang. 2026-08-22. Dual $L_p$ John ellipsoids for general measures. https://arxiv.org/abs/2608.21958

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