### Total diameter and area of closed submanifolds

- Series
- Geometry Topology Seminar
- Time
- Monday, December 2, 2013 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Mohammad Ghomi – Georgia Tech – ghomi@math.gatech.edu

The total diameter of a closed planar curve C 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.