  In this note we introduce a natural Finsler structure on convex surfaces,
referred to as the projective Finsler structure, which is dual in a sense to
the obvious inclusion of a convex surface in a normed space. It has an
associated projective girth, which is similar to the notion of girth defined by
Sch\"affer. We prove the analogs of Sch\"affer's dual girth conjecture (proved
by \'Alvarez-Paiva) and the Holmes-Thompson dual volumes theorem in the
projective setting. We then show that the projective Finsler structure admits a
natural extension to higher Grassmannians, and prove the corresponding theorems
in the general case. We follow \'Alvarez-Paiva's approach to the problem,
namely, we study the symplectic geometry of the associated co-ball bundles. For
the higher Grassmannians, the theory of Hamiltonian actions is applied.
