For a finite dimensional Banach space $V$ with strictly convex and smooth norm (outside the origin), there is a natural way to induce a Riemannian structure on the unit sphere $S$ (see Alvarez-Paiva, "Some problems on Finsler Geometry"): consider $L=\|x\|^2/2 $, then $dL$ is a diffeomorphism from $V\backslash0$ to its dual $V^*\backslash0$. Now, for $q\in S$ we get $D_q(dL):T_qV \rightarrow T_qV^*$, and for $u,v\in T_qS$ take $g_q(u,v)=D_q(dL)(u)(v)$. What is known about this structure? What are the geodesics, for instance? What are their lengths?
Maybe i'm wrong, but isn't $D_q(dL)$ the legendre transform of of the "Lagrangian"? In which case you should get the usual metric back. See for example Sternberg's book on differential geometry (the chapter on calculus of variations) or (i know, it sounds strange but) Mackey's Mathematical Foundations of Quantum Mechanics (the first chapter). If my memory doesn't fail me Abraham, Marsden should have something on that specific example too.
Edit: This applies in the common case where the norm is induced by an inner product.