  We construct a convex and strongly pseudoconvex Kobayashi positive Finsler
metric on a vector bundle $E$ under the assumption that the symmetric power of
the dual $S^kE^*$ has a Griffiths negative $L^2$-metric for some $k$. The proof
relies on the negativity of direct image bundles and the Minkowski inequality
for norms. As a corollary, we show that given a strongly pseudoconvex Kobayashi
positive Finsler metric, one can upgrade to a \textit{convex} Finsler metric
with the same property. We also give an extremal characterization of Kobayashi
curvature for Finsler metrics.
