Positively curved Finsler metrics on vector bundles

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 SkE∗SkE∗ has a Griffiths negative L2L2 -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 convex Finsler metric with the same property. We also give an extremal characterization of Kobayashi curvature for Finsler metrics.