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Article overview
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On the sine polarity and the $L_p$-sine Blaschke-Santal'{o} inequality | | Qingzhong Huang
; Ai-Jun 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 Blaschke-Santal’{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 Blaschke-Santal’{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 $L_p$ Blaschke-Santal’{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 Blaschke-Santal’{o}
inequality reduces to the sine Blaschke-Santal’{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 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-Santal’{o} inequality are settled. | | Source: | arXiv, 2206.00185 | | Services: | Forum | Review | PDF | Favorites | |
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