Convex Body Bonnesen Aleksandrov-fenchel Inequality

The classical isoperimetric inequality, the Bonnesen-style inequality and the Aleksandrov-Fenchel inequality are important inequalities in geometry. The isoperi-metric deficit measures the deficit between a geometric body and a disc. Had-wiger, Osserman, Klain, Bottema and Zhou and so on make efforts to investigate the lower bound and the upper bound of the isoperimetric deficit. In this paper, we mainly study two problems. Firstly, we investigate the Bonnesen-style inequal-ity of convex domain in a plane of constant curvature, and obtain analogues of the Bonnesen inequality that strengthen the known Bonnesen style inequalities. Secondly, using the known Aleksandrov-Fenchel inequality, we study the sym-metric mixed homothetic deficit△i(K, L) of convex bodies K and L in Rn and the Aleksandrov-Fenchel deficit△i(K). Moreover the Aleksandrov-Fenchel deficit is the generation of isoperimetric deficit, the Bonnesen Aleksandrov-Fenchel in-equality (the lower bound of Aleksandrov-Fenchel deficit) is the generation of Bonnesen inequality.The isoperimetric deficit of K in a plane of constant curvature is△∈(K)=L2-4πA+∈A2(where K is the domain of area A and circum-length L,∈is the curvature of the plane of constant curvature), Klain, Zhou and Chen obtain sev-eral lower bounds of the isoperimetric deficit in a plane of constant curvature by differential methods. In this paper we first investigate the containment measure idea and the fundamental kinematic formula in integral geometry, and obtain two lower bounds of the isoperimetric deficit that strengthen the known isoperimetric deficit. We have the following theorems:Theorem3.3.9Let K be a strictly convex domain of area A and circum-length L in X∈2, then where re, ri are the smallest geodesic circumradius and the biggest geodesic in-radius of K respectively. The equality sign holds if and only if K is a geodesic disc. Theorem3.3.11Let K be a. strictly convex domain of area A and circum-length L in X∈2, then where re, ri are the smallest geodesic circumradius and the biggest geodesic in-radius of K respectively. The equality sign holds if and only if K is a geodesic disc.The famous Aleksandrov-Fenchel inequality is very important in Brunn-Minkowski theory. Except the classical Aleksandrov-Fenchel inequality, Lutwak. Yang, Zhang obtain several different forms of Aleksandrov-Fenchel inequality. Us-ing the classical Aleksandrov-Fenchel inequality, we study the symmetric mixed homothetic deficit△i(K, L)(where△i(K, L)=Vi2(K, L)-Vi-1(K, L)Vi+1(K, L) is the mixed volum of K, L) and the Aleksandrov-Fenchel isoperimetric deficit△i(K)=Wi2(K)-Wi-1(K)Wi+1(K)(where Wi(K) i-th quermassintegral of K), and obtain several lower bounds of the isoperimetric deficit. As direct corollar-ies, we obtain the lower bounds of isoperimetric deficit expressed by the width function of project body in two-dimensional, three-dimensional Euclidean space. Finally, we obtain an upper bounds of isoperimetric deficit. The results are:Theorem4.2.10Let K be a convex body in Rn and Wi(K) i-th quer-massintegral of K. Then where u∈Sn-1, u⊥is the linear subspace perpendicular to u, Ku is the orthogonal projection of a convex body K onto the linear subspace u⊥, ωd-1(Ku) is the mean width of Ku, and ωmax(Ku) and ωmin(Ku) are the maximum and minimum width of ω(Ku) over all u∈Sn-1.Theorem4.3.2Let K be a convex body in Rn and Wi(K) i-th quermass-integral of K. If1≤i≤n-1, then△i(K)=Wi2(K)-Wi+1(K)Wi-1(K)≤Wi+1(K)Wi(K)(R(K, B)-r(K, B)), where r(K, B) and R(K, B) are the relative inradius and the relative circumradius of K with respect to B, respectively.