两平面凸域的对称混合等周不等式

曾春娜, 周家足, 岳双珊

数学学报 ›› 2012 ›› Issue (2) : 355-362.

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数学学报 ›› 2012 ›› Issue (2) : 355-362. DOI: 10.12386/A2012sxxb0035
论文

两平面凸域的对称混合等周不等式

    曾春娜1, 周家足1,2, 岳双珊1
作者信息 +

The Symmetric Mixed Isoperimetric Inequality of Two Planar Convex Domains

    Chun Na ZENG1, Jia Zu ZHOU1,2, Shuang Shan YUE1
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摘要

设 Kk(k=#em/em#,j) 为欧氏平面 R2 中面积为 Ak,周长为 Pk 的域,它们的对称混合等周亏格(symmetric mixed isoperimetric deficit) 为 σ(K#em/em#,Kj)=P#em/em#2Pj2-16π2A#em/em#Aj.根据周家足,任德麟 (2010)和Zhou,Yue(2009)中的思想,用积分几何方法,得到了两平面凸域的Bonnesen型对称混合不等式及对称混合等周不等式,给出了两域的对称混合等周亏格的一个上界估计.还得到了两平面凸域的离散Bonnesen型对称混合不等式及两凸域的对称混合等周亏格的一个上界估计,并应用这些对称混合(等周)不等式估计第二类完全椭圆积分.

Abstract

Let Kk(k=#em/em#,j) be domain of the area Ak, and of the perimeter Pk, respectively. The symmetric mixed isoperimetric deficit of K#em/em# and Kj is defined as σ(K#em/em#,Kj)=P#em/em#2Pj2-16π2A#em/em#Aj. We follow the ideas of Zhou, Ren (2010) and Zhou, Yue (2009) and obtain some Bonnesen-style symmetric mixed inequalities and the symmetric mixed isoperimetric inequality by the method of integral geometry. We also obtain some symmetric mixed isoperimetric upper limits. Some discrete Bonnesen-style symmetric mixed inequalities and one upper limit of the discrete symmetric mixed isoperimetric deficit for two domains are obtained. Finally we apply these symmetric mixed (isoperimetric) inequalities to estimate the complete elliptic integral of second class.

关键词

对称混合等周亏格 / 对称混合等周不等式 / Bonnesen型对称混合不等式

Key words

The symmetric mixed isoperimetric deficit / the symmetric mixed isoperimetric inequality / the Bonnesen-style symmetric mixed inequality

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曾春娜, 周家足, 岳双珊. 两平面凸域的对称混合等周不等式. 数学学报, 2012(2): 355-362 https://doi.org/10.12386/A2012sxxb0035
Chun Na ZENG, Jia Zu ZHOU, Shuang Shan YUE. The Symmetric Mixed Isoperimetric Inequality of Two Planar Convex Domains. Acta Mathematica Sinica, Chinese Series, 2012(2): 355-362 https://doi.org/10.12386/A2012sxxb0035

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基金

国家自然科学基金资助项目(10971167)

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