Some Remarks On Inclusion Measure

In this paper, we mainly investigate some property of the inclusion measure, and obtain a sufficient condition about a convex body contains another, that is the generalize of Hadwiger condition in Rn. Especially in R4 and R5, we get a better conclusion. Finally, we get a better upper and lower limits of the complete elliptic integral of the second kind, through the plane isoperimetric inequality and the Bonnesen-style inequalities.We get the following theorems:Theorem 2.5. Suppose K and L are convex bodies in Rn, we haveifα＞β＞0, we have C(αK,βL,1)(?)αC(K, L,1);ifβ＞α＞0, we have C(αK,βL,1)(?)αC(K, L,1).Theorem 2.6. Suppose K and L are convex bodies in Rn,, ifα＞0, we haveTheorem 2.7. Suppose K and L are convex bodies in Rn,ifα＞β＞0, we have m(βL (?)αK)＞αnm(L (?) K);ifβ＞α＞0, we have m(βL (?)αK)＜αnm(L (?) K).Theorem 2.8. Suppose K1 and K2 are convex bodies in Rn, K1+K2 is their Minkowski addition, we have Theorem 2.9. Suppose K1 and K2 are convex bodies in Rn, we haveTheorem 2.11. Suppose Li:i=1,2,…, n and K are convex bodies in Rn we haveTheorem 2.13. Suppose K and L are convex bodies in Rn(n≥3), V(K) and A(K) are the volume and area of K, V(L) and M(L) are the volume and the mean width of L, if V(K)≥V(L), we have equation holds if and only if K and L are balls.Theorem 2.14. Suppose K and L are convex bodies in Rn(n≥3), V(K) and A(K) are the volume and area of K, V(L) and M(L) are the volume and the mean width of L, if V(K)≥V(L), then the following states that K can contain L:Theorem 2.15. Suppose K and L are convex bodies in Rn(n≥3), V(K) and A(K) are the volume and area of K, V(L) and M(L) are the volume and the mean width of L, if V(K)≥V(L), we haveCorollary 2.16. Suppose K and L are convex bodies in R3, S(K) and V(K) are area and volume of K, V(L) and M(L) are volume and the mean width of L, if if V(K)≥V(L), we have Theorem 2.18. Suppose K and L are convex bodies in R4, V(K) and A(K) are the volume and area of K, V(L) and M(L) are the volume and the mean width of L, if V(K)≥V(L), we have this is a sufficient condition that K can contain L.Theorem 2.19. Suppose K and L are convex bodies in R5, V(K) and A(K) are the volume and area of K, V(L) and M(L) are the volume and the mean width of L, if V(K)≥V(L), we have this is a sufficient condition that K can contain L.Theorem ??. Suppose D(a,b) is elliptic, b≤r≤a, we have equation holds if and only if a=b,that is D is circle.