R^3中紧致凸曲面的等周不等式

On the Isoperimetric Inequality for Compact Convex Surface in R^3

∑是R3中紧致凸曲面,∑上有一定长为L的简单闭曲线C,由C围成的区域D的面积为A,利用ΔΣr2=4+4H〈r,n〉、散度定理和庞加莱不等式,得等周不等式为L2≥4πA+4πp0H0,其中H0=min｛Hp,Hp为Σ在p点的平均曲率｝,p0=max｛〈r,n〉p,〈r,n〉p为Σ在p点的支撑函数的值｝。

Let ∑ be a compact convex surface in R^3. A simple closed C curve C of length L encloses damain D of area A on ∑. Using A∑r^2=4＋4H（r,n）, the divergence theorem and the PoinearE inequality, isoperimetric inequality L^2≥4πA ＋4πp0H0 is gotten. Where H0=min {Hp,Hp is the mean curture at point p }, p0=max {（r,n）p, 〈r,n）p is the support function at point p }.