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Motivated by some aspect of large-scale symplectic topology, we prove thatfor any pair $g_0, \, g_1$ of complete Riemannian metrics of bounded curvatureand of injectivity radius bounded away from zero, the convex sum $g_s: = (1-s )g_0 + s g_1$ also has bounded curvature depending only on the curvature bounds$\|R_{g_i}\|_{C^0}$ of $g_0$ or $g_1$, and that the injectivity radii of $g_s$have uniform lower bound depending only on the derivative bounds$\|R_{g_i}\|_{C^1} = \|R_{g_i}\|_{C^0} + \|DR_{g_i}\|_{C^0}$. A main technicalingredient to establish the injectivity radius lower bound is an application ofthe quantitative inverse function theorem. This proves that each quasi-isometryclass of tame metrics is convex and so contractible in (weak) $C^\infty$topology. The main result of the present paper is a key ingredient entering inthe construction of large-scale symplectic topological invariants, via theanalysis of pseudoholomorphic curves on open symplectic manifolds, whichdepends only on the quasi-isometry class of tame almost complex structures.

Injectivity radius lower bound of convex sum of tame Riemannian metrics and its applications