  
  
Stat 
Members: 3657 Articles: 2'599'751 Articles rated: 2609
06 October 2024 

   

Article overview
 

On the sine polarity and the $L_p$sine BlaschkeSantal'{o} inequality  Qingzhong Huang
; AiJun Li
; Dongmeng Xi
; Deping Ye
;  Date: 
1 Jun 2022  Abstract:  This paper is dedicated to study the sine version of polar bodies and
establish the $L_p$sine BlaschkeSantal’{o} inequality for the $L_p$sine
centroid body.
The $L_p$sine centroid body $Lambda_p K$ for a star body
$Ksubsetmathbb{R}^n$ is a convex body based on the $L_p$sine transform, and
its associated BlaschkeSantal’{o} inequality provides an upper bound for the
volume of $Lambda_p^{circ}K$, the polar body of $Lambda_p K$, in terms of
the volume of $K$. Thus, this inequality can be viewed as the "sine cousin" of
the $L_p$ BlaschkeSantal’{o} inequality established by Lutwak and Zhang. As
$p
ightarrow infty$, the limit of $Lambda_p^{circ} K$ becomes the sine
polar body $K^{diamond}$ and hence the $L_p$sine BlaschkeSantal’{o}
inequality reduces to the sine BlaschkeSantal’{o} 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 originsymmetric convex bodies
generated by the intersection of originsymmetric 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 BlaschkeSantal’{o} inequality are settled.  Source:  arXiv, 2206.00185  Services:  Forum  Review  PDF  Favorites 


No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.

 


