Total diameter and area of closed submanifolds

Abstract

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