On the monotone properties of general affine surfaces under the Steiner symmetrization
On the monotone properties of general affine surfaces under the Steiner symmetrization
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DOI：
10.1512/iumj.2014.63.5205
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发表时间：
2012-05
影响因子：
1.1
通讯作者：
Deping Ye
中科院分区：
文献类型：
--
作者：
Deping Ye
本文证明，如果函数(凹)$\phi$和(凸)$\psi$满足一定条件，$L_{\phi}$仿射表面积是单调递增的，而$L_{\psi}$仿射表面积在Steiner对称化下是单调递减的。因此，我们可以在 $\phi$ 和 $\psi$ 上的某些条件下证明相关的仿射等周不等式，而无需假设所涉及的凸体在原点具有质心（或 Santal\'{o} 点）。
In this paper, we prove that, if functions (concave) $\phi$ and (convex) $\psi$ satisfy certain conditions, the $L_{\phi}$ affine surface area is monotone increasing, while the $L_{\psi}$ affine surface area is monotone decreasing under the Steiner symmetrization. Consequently, we can prove related affine isoperimetric inequalities, under certain conditions on $\phi$ and $\psi$, without assuming that the convex body involved has centroid (or the Santal\'{o} point) at the origin.