Research About The Mean Curvature Integral Of A Hypersurface In R~n And The Complete Ellipite Integral Of The Second Kind

In this article, we mainly investigate three problems. First of all, According to a sufficient condition for one convex body containing another in the Euclidean space R3, we have the lower limits of the integral of the square of the mean curvature∫(?)D H2da about boundary surface (?)D of a convex domain D; Secondly, we have an inequality of the integral of the mean curvature about the hypersurface in the Euclidean space Rn;Finally, we estimate some upper and lower limits of the com-plete ellipitic integral of the second kind via the geometry equalities (isopermetric inequality, the symmetric mixed isopermetric inequality, Bonnesen-type symmetric mixed isopermetric inequality).we have the following theorems:Theorem 2.2. Let D be a convex body with the class C2-smooth boundary surface (?)D in the Euclidean space R3. Let V be the volume of D, H, A, respectively, be the mean curvature, the area of (?)D, then we have with equality if D is a ball.Theorem 2.3. Let D be a convex body with the class C2-smooth boundary surface (?)D in the Euclidean space R3. Let V be the volume of D, H, A, respectively, be the mean curvature, the area of (?)D. Let ri and re be, respectively, the radiuses of the biggest inscribed ball and the smallest circumscribed ball of D, then for any ball of radius r(ri≤r≤re), we have with equality if D is a ball of radius r.Theorem 2.4. Let D be a convex body with the class C2-smooth boundary surface (?)D in the Euclidean space R3. Let V be the volume of D, H, A, respectively, be the mean curvature, the area of (?)D. Let ri and re be, respectively, the radiuses of the biggest inscribed ball and the smallest circumscribed ball of D, then we have with equality if ri=re, that is, if D is a ball.Theorem 3.3. Let∑be a compact embedded hypersurface of the class C2-smooth, bounding a domain of volume V, A be the area of∑. If the mean curvature H of∑is positive everywhere, then where dσis the area element of E. Equality holds if and only if∑is a hypersphere.Theorem 4.18. Let a and b be respectively, the long axle length and the short axle length of the ellipse E(a, b) in the Euclidean plane R2, then the complete elliptic integral of the second kind E(e) satisfies where1/a≤t≤1/b Each equality holds if and only if a=b, that is, if the ellipse E(a, b) is a disc.Theorem 4.21. Let a and b be respectively, the long axle length and the short axle length of the ellipse E(a,b) in the Euclidean plane R2, then the complete elliptic integral of the second kind E(e) satisfies Each equality holds if and only if a=b, that is, if the ellipse E(a, b) is a disc.Theorem 4.22. Let a and b be respectively, the long axle length and the short axle length of the ellipse E(a, b) in the Euclidean plane R2, then the complete elliptic integral of the second kind E(e) satisfies Equality holds if and only if a=b, that is, if the ellipse E(a, b) is a disc.Theorem 4.23. Let a and b be respectively, the long axle length and the short axle length of the ellipse E(a, b) in the Euclidean plane R2, then the complete elliptic integral of the second kind E(e) satisfies Equality holds if and only if a=b, that is, if the ellipse E(a, b) is a disc.Theorem 4.24. Let a and b be respectively, the long axle length and the short axle length of the ellipse E(a,b) in R2, then the complete elliptic integral of the second kind E(e) satisfies Equality holds if and only if a=b, that is, if the ellipse E(a, b) is a disc.