arXiv subjects

Qingzhong Huang

Publications and source records attributed to Qingzhong Huang.

6 recordsLinked to original sources

arXiv2026

Dual $L_p$ John ellipsoids for general measures

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.

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↗

arXiv2022

Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at $s=1$ and $s=0$

In 2014, Ludwig showed the limiting behavior of the anisotropic Gagliardo $s$-seminorm of a function $f$ as $s\rightarrow 1^-$ and $s\rightarrow0^+$, which extend the results due to Bourgain-Brezis-Mironescu(BBM) and Maz'ya-Shaposhnikova(MS) respectively. Recently, Brezis, Van Schaftingen and Yung provided a different approach by replacing the strong $L^p$ norm in the Gagliardo $s$-seminorm by the weak $L^p$ quasinorm. They characterized the case for $s=1$ that complements the BBM formula. The corresponding MS formula for $s=0$ was later established by Yung and the first author. In this paper, we follow the approach of Brezis-Van Schaftingen-Yung and show the anisotropic versions of $s=1$ and $s=0$. Our result generalizes the work by Brezis, Van Schaftingen, Yung and the first author and complements the work by Ludwig.

math.FA↗

arXiv2021

On the Musielak-Orlicz-Gauss image problem

In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body $K$, its Musielak-Orlicz-Gauss image measure, denoted by $\widetilde{C}_Θ(K, \cdot)$, involves a triple $Θ=(G, Ψ, λ)$ where $G$ and $Ψ$ are two Musielak-Orlicz functions defined on $S^{n-1}\times (0, \infty)$ and $λ$ is a nonzero finite Lebesgue measure on the unit sphere $S^{n-1}$. Such a measure can be produced by a variational formula of $\widetilde{V}_{G, λ}(K)$ (the general dual volume of $K$ with respect to $λ$) under the perturbations of $K$ by the Musielak-Orlicz addition defined via the function $Ψ$. The Musielak-Orlicz-Gauss image problem contains many intensively studied Minkowski type problems and the recent Gauss image problem as its special cases. Under the condition that $G$ is decreasing on its second variable, the existence of solutions to this problem is established.

math.MG↗

arXiv2016

Linear absorption coefficient of in-plane graphene on a silicon microring resonator

We demonstrate that linear absorption coefficient (LAC) of a graphene-silicon hybrid waveguide (GSHW) is determined by the optical transmission spectra of a graphene coated symmetrically coupled add-drop silicon microring resonator (SC-ADSMR), of which the value is around 0.23 dB/um. In contrast to the traditional cut-back method, the measured results are not dependent on the coupling efficiency of the fiber tip and the waveguide. Moreover, precision evaluation of graphene coated silicon microring resonator (SMR) is crucial for the optoelectronic devices targeting for compact footprint and low power consumption.

physics.optics↗