一般测度的对偶$L_p$ John椭球
Dual $L_p$ John ellipsoids for general measures
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中文总结 AI 辅助
本文在对偶加权$L_p$ Brunn-Minkowski理论框架下，将对偶$L_p$ John椭球推广到含高斯测度的一般测度情形，证明了对应优化问题解的相关性质并给出相关几何结果。
中文摘要 AI 辅助
对偶$L_p$ John椭球包含经典的Löwner椭球和Legendre椭球，它是某类优化问题的解。本文在对偶加权$L_p$ Brunn-Minkowski理论框架下考虑了广泛的推广情形，建立了具有局部可积密度的一般测度在$L_p$-调和径向组合下的变分公式，由此提出了针对一般测度的对应优化问题。我们证明了该问题解的存在性、唯一性与特征，该解定义了一类新型椭球，将对偶$L_p$ John椭球推广到更广泛的情形，包括高斯测度，同时给出了相关几何不等式与一般测度的$L_p$ Löwner包含关系。
英文摘要
The dual $L_p$ John ellipsoid, including the classical Löwner 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 $L_p$ Löwner inclusion for general measures are also given.
