The Shortest Geodesic Flower on a Manifold with Locally Convex Ends and Finite Volume

Suppose $M$ is a complete, non-compact $n$-dimensional Riemannian manifold with locally convex ends and finite volume. We prove that $M$ admits a non-trivial geodesic net with one vertex, at most $(n+2)(n+1)\mathrm{/}2$ edges, and total length at most $$(n+2)(n+1)(n\mathrm{/}2)\mathrm{vol}(M{)}^{1\mathrm{/}n}\mathrm{.}$$ This is a quantitative version of a result of G. R. Chambers, Y. Liokumovich, A. Nabutovsky and R. Rotman.