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It is shown that if the Kato constant of the negative part of the Riccicurvature below a positive level is small, then the volume of the correspondingmanifold can be bounded above in terms of the Kato constant and the total Riccicurvature. Together with the results from [5] and [6], this yields ageneralization of the famous Bonnet-Myers theorem. Connections to some earliergeneralizations are discussed.

Almost positive Ricci curvature in Kato sense -- an extension of Myers' theorem