Strictly convex norms instead of inner products on the tangent spaces On differentiable manifolds, is there a term for functions $f$ from the tangent bundle to the real line such that for all points of the manifold, the restriction of $f$ to that point's tangent space is a strictly convex norm and 
 the restriction of $ f$ to the complement of the zero subbundle is smooth? 
Is there a term for a manifold together with such a function?

Strictly convex Finsler metric or strictly convex Finsler structure. Usage examples. Unfortunately, "strictly convex Finsler manifold" is likely to be confused with geodesic convexity of a manifold; which is probably why this particular combination did not   come up in search.
If you don't insist on the "strictly convex" part, then it's just a Finsler manifold.
