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In this paper, we mainly establish a Cheeger type finiteness theorem forBerwald manifolds. In order to do this, we study the injectivity radius and theconvex radius of a Finsler manifold. A Cheeger type estimate on injectivityradii for Finsler manifolds is given and the existence of the center of mass ofa Berwald manifold is proved.

A Cheeger type finiteness theorem for Finsler manifolds