In the seventies, Schaeffer conjectured that the girth of a normed space --- the infimum of the lengths of all curves on its unit sphere that join a pair of antipodes --- equals the girth of its dual. In this talk I will present a "book proof" using elementary symplectic geometry. It is also shown that other results in convex geometry such as Crofton's formula for hypersurfaces in finite-dimensional spaces and the convexity of projection bodies are also consequences of the same symplectic result that implies Schaeffer's conjecture.