Mean curvature flow and isoperimetric inequalities
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Then we consider the inequality of the form.
The inequality of type 9 is called the Bonnesen-style Wulff isoperimetric inequality. Its reverse form, that is,. Each equality holds if and only if K and W are homothetic. The support function p K u of the convex body K is defined by.
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For simplicity, we replace p K u by p K. Let p K be the support function of K , then. Inequality 22 holds as an equality if and only if. By 17 , we have. Each inequality in 25 holds as an equality if and only if K and W are homothetic.
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This contradicts the maximality of r W. Inequality 26 is strict. For general convex bodies, Luo, Xu and Zhou [ 17 ] have also obtained inequality 26 by the integral geometry method. However, it is difficult to obtain the equality condition of inequality 26 for general convex bodies.
Via the method of convex geometric analysis, a complete proof of inequality 26 with equality condition is given in [ 9 ]. We obtain the following Wulff isoperimetric inequality.
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By inequalities 28 , 29 , we have, respectively,. Then inequalities 30 , 31 can be, respectively, rewritten as. This proves inequality Inequalities 30 , 31 can also be rewritten, respectively, as follows:. Inequality 12 is proved. Therefore we have the following. Let r and R be , respectively , the radius of the maximum inscribed disc and the radius of the minimum circumscribed disc of K. It should be noted that 32 is obtained in [ 24 ], which is stronger than the Bonnesen isoperimetric inequality 4.
Each inequality holds as an equality if and only if K and W are homothetic. Recalling 6 , 18 and 19 , we have. By the definition of L K , W in 6 , we have. By 36 , 37 and 38 , we have. Inequalities 38 , 37 can, respectively, be rewritten as. Together with 36 and the above inequalities, inequalities 35 follow. By inequalities 34 , we have. This is inequality By 39 and 40 , we obtain.
According to the equality conditions of 39 and 40 , the equality holds for 15 if and only if K and W are homothetic. This gives inequality By 39 and 40 again, we get. From the equality conditions of 39 and 40 again, the equality of 16 holds if and only if K and W are homothetic. Let W be a unit disc. The authors would like to thank anonymous referees for helpful comments and suggestions that directly led to the improvement of the original manuscript.
Publisher: Birkhauser Basel. See details.
See all 2 brand new listings. Buy It Now. Add to cart. Be the first to write a review About this product. About this product Product Information Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized.
The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds.
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One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows.
Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds. The following ISBNs are associated with this title:.
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