  This paper is devoted to investigate an interpolation inequality between the
Brezis-V\'azquez and Poincar\'e inequalities (shortly, BPV inequality) on
nonnegatively curved spaces. As a model case, we first prove that the BPV
inequality holds on any Minkowski space, by fully characterizing the existence
and shape of its extremals. We then prove that if a complete Finsler manifold
with nonnegative Ricci curvature supports the BPV inequality, then its flag
curvature is identically zero. In particular, we deduce that a Berwald space of
nonnegative Ricci curvature supports the BPV inequality if and only if it is
isometric to a Minkowski space. Our arguments explore fine properties of Bessel
functions, comparison principles, and anisotropic symmetrization on Minkowski
spaces. As an application, we characterize the existence of nonzero solutions
for a quasilinear PDE involving the Finsler-Laplace operator and a Hardy-type
singularity on Minkowski spaces where the sharp BPV inequality plays a crucial
role. The results are also new in the Riemannian/Euclidean setting.
