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Dual Lp John ellipsoids for general measures

  • Ai-Jun Li
  • Baihe Liu
  • Qingzhong Huang
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Abstract

The dual Lp 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 Lp 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 Lp-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 Lp 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 Lp Löwner inclusion.

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Cite

BibTeXRIS
BibTeX
@misc{li2026dual,
  title = {{Dual $L_p$ John ellipsoids for general measures}},
  author = {Ai-Jun Li and Baihe Liu and Qingzhong Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.21958},
  eprint = {2608.21958},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.21958},
}
RIS
TY  - UNPB
TI  - Dual L_p John ellipsoids for general measures
AU  - Ai-Jun Li
AU  - Baihe Liu
AU  - Qingzhong Huang
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2608.21958
UR  - https://arxiv.org/abs/2608.21958
AB  - 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.
ER  -