  In Minkowski geometry the metric features are based on a compact convex body
containing the origin in its interior. This body works as a unit ball with its
boundary formed by the unit vectors. Using one-homogeneous extension we have a
so-called Minkowski functional to measure the lenght of vectors. The half of
its square is called the energy function. Under some regularity conditions we
can introduce an average Euclidean inner product by integrating the Hessian
matrix of the energy function on the Minkowskian unit sphere. Changing the
origin in the interior of the body we have a collection of Minkowskian unit
balls together with Minkowski functionals depending on the base points. It is a
kind of special Finsler manifolds called a Funk space. Using the previous
method we can associate a Riemannian metric as the collections of the Euclidean
inner products belonging to different base points. We investigate this
procedure in case of Finsler manifolds in general. Central objects of the
associated Riemannian structure will be expressed in terms of the canonical
data of the Finsler space. We take one more step forward by introducing
associated Randers spaces by averaging of the vertical derivatives of the
Finslerian fundamental function. Such kind of perturbation will have a crucial
role when we apply the general results to Funk spaces together with some
contributions to Brickell's conjecture on Finsler manifolds with vanishing
curvature tensor $\mathbb{Q}$.
