A solution to a conjecture of Schaeffer. Posted in the Finsler Geom. In the 1970's, Schaeffer defined the girth of a normed space as the infimum of the lengths of all curves on the unit sphere that join a pair of antipodes. He conjectured that the girth of a normed space equals the girth of its dual. This note is an annoucement of the proof of this conjecture.
Dual spheres have the same girth.
In this paper, I use some simple symplectic geometry to study the inner geometry of unit spheres in normed spaces. It includes the full proof of the girth conjecture as well as new proofs of some results of Holmes, Thompson, and El-Ekhtiar. Tautness is invariant under Lie sphere transformations. In 1987, Cecil and Chern showed that tautness is invariant under Lie sphere transformations. This note presents a very simple proof of this result and a manifestly invariant definition of tautness for Legendrian submanifolds in the space of cooriented contact elements of the sphere.