论文标题
在正弦和$ L_P $ -Sine Blaschke-Santaló不平等上
On the sine polarity and the $L_p$-sine Blaschke-Santaló inequality
论文作者
论文摘要
本文致力于研究Polar Body的正弦版本，并为$ L_P $ -Sine Centroid身体建立$ L_P $ -Sine Blaschke-Santaló不平等。 $ l_p $ -sine Centroid身体$λ_pk $用于星体$ k \ subset \ subset \ mathbb {r}^n $是一个基于$ l_p $ -Sine变换的凸形主体，其相关的blaschke-santaló不平等为$λ_p^$ compor的体积提供了$λ_p^$ compor的bossantaló不平等， $ k $。因此，这种不平等可以被视为Lutwak和Zhang确定的$ L_P $ blaschke-Santaló不平等的“正弦表亲”。作为$ p \ rightarrow \ infty $，$λ_p^{\ circ} k $的极限变成了正弦物体$ k^{\ diamond} $，因此$ l_p $ -sine blaschkeke-santaló不平等降低了Sine blaschke-santaló的Sine blaschke-santaló，对Sine Polar Polars sine Polar polare sine polar polare sine sine polar to sine polar sine polar bementality sine blaschke-santaló。正弦极性自然会导致一类新的凸体$ \ MATHCAL {C} _ {E}^n $，由所有原点对称凸形组成，由原点对称性封闭的固体圆柱体的交汇处产生。开发了$ \ Mathcal {C} _ {E}^n $中的许多概念，包括圆柱支撑功能，支撑缸，圆柱形高斯图像和圆柱船体。基于这些新引入的概念，正弦blaschke-santaló不平等的平等条件得到了解决。
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.
