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Let   M  (  n  ,  D  )   be the space of closed n -dimensional Riemannian manifolds ( M ,  g ) with   diam  (  M  )  ≤  D   and    |    sec  M    |  ≤  1    . In this paper we consider sequences   (   M  i   ,   g  i   )   in   M  (  n  ,  D  )   converging in the Gromov-Hausdorff topology to a compact metric space Y . We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient    vol  (   B  r   M  i(  x  )   )inj    M  i(  x  )can be uniformly bounded from below by a positive constant C ( n ,  r ,  Y ) for all points   x  ∈   M  i    . On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C ( n ,  r ,  Y ) bounding the quotient    vol  (   B  r   M  i(  x  )   )inj    M  i(  x  )uniformly from below for all   x  ∈   M  i    . As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in   M  (  n  ,  D  )   with   C  ≤    vol  (  M  )    inj  (  M  ).

A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter