Total Diameter and Area of Closed Submanifolds
نویسنده
چکیده
The total diameter of a closed planar curve C ⊂ R is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of C. Furthermore, when C is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when C is a circle. We also generalize these results to m dimensional submanifolds of R, where the “area” will be defined in terms of the mod 2 winding numbers of the submanifold about the n−m− 1 dimensional affine subspaces of R.
منابع مشابه
relating diameter and mean curvature for submanifolds of euclidean space
Given a closed m-dimensional manifold M immersed in R, we estimate its diameter d in terms of its mean curvature H by
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