arXiv subjects

Ai-Jun Li

Publications and source records attributed to Ai-Jun Li.

3 recordsLinked to original sources

arXiv2026

Dual $L_p$ John ellipsoids for general measures

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.

math.MG↗

arXiv2022

Mean width inequalities of sections and projections for isotropic measures

In this paper, we establish mean width inequalities of sections and projections of convex bodies for isotropic measures with complete equality conditions, which extends the recent work of Alonso-Gutiérrez and Brazitikos. Different from their approach, our proof is based on the approach developed by Lutwak, Yang and Zhang, by using the Ball-Barthe inequality, the mass transportation, and the isotropic embedding.

math.MG↗

arXiv2022

On the sine polarity and the $L_p$-sine Blaschke-Santaló inequality

This paper is dedicated to study the sine version of polar bodies and establish the $L_p$-sine Blaschke-Santaló inequality for the $L_p$-sine centroid body. The $L_p$-sine centroid body $Λ_p K$ for a star body $K\subset\mathbb{R}^n$ is a convex body based on the $L_p$-sine transform, and its associated Blaschke-Santaló inequality provides an upper bound for the volume of $Λ_p^{\circ}K$, the polar body of $Λ_p K$, in terms of the volume of $K$. Thus, this inequality can be viewed as the "sine cousin" of the $L_p$ Blaschke-Santaló inequality established by Lutwak and Zhang. As $p\rightarrow \infty$, the limit of $Λ_p^{\circ} K$ becomes the sine polar body $K^{\diamond}$ and hence the $L_p$-sine Blaschke-Santaló inequality reduces to the sine Blaschke-Santaló inequality for the sine polar body. The sine polarity naturally leads to a new class of convex bodies $\mathcal{C}_{e}^n$, which consists of all origin-symmetric convex bodies generated by the intersection of origin-symmetric closed solid cylinders. Many notions in $\mathcal{C}_{e}^n$ are developed, including the cylindrical support function, the supporting cylinder, the cylindrical Gauss image, and the cylindrical hull. Based on these newly introduced notions, the equality conditions of the sine Blaschke-Santaló inequality are settled.

math.MG↗