2013/12/04 by Mohammad Ghomi, Ralph Howard, Ghomi, Mohammad +1
Mathematics · #52A20 #53A07 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG) #math.DG #math.GT #math.MG #msc:52A20 #msc:53A07
paper · pdf · doi:10.48550/arxiv.1312.1384
13 pages, 4 figures; Minor revisions; Accepted for publication in Mathematische Annalen
arxiv created 2015/01/18 · arxiv updated 2015/01/20
The total diameter of a closed planar curve C⊂ R2 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 Rn, 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 Rn.