Total diameter and area of closed submanifolds

Abstract

The total diameter of a closed planar curve $C\subset R^2$ 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 $R^n$, 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 $R^n$.