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?