两平面凸域的对称混合等周不等式

The Symmetric Mixed Isoperimetric Inequality of Two Planar Convex Domains

设K_k(k=i,j)为欧氏平面R^2中面积为A_k,周长为P_k的域,它们的对称混合等周亏格(symmetric mixed isoperimetric deficit)为σ(K_i,K_j)=P_i^2P_j^2-16π~2A_iA_j.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.

Let Kk （k = i,j） be domain of the area Ak, and of the perimeter Pk, respectively. The symmetric mixed isoperimetric deficit of h＇i and Kj is defined σ（Ki,Kj）=Pi^2Pj^2-16π^2AiAj. We follow the ideas of Zhou, Ren （2010） and Zhou, Yue （2009） and obtain some Bonnesen-style symmetric mixed inequalities and the sym- metric mixed isoperimetric inequality by the method of integral geometry. We also ob- tain 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.