A splitting theorem for nonnegatively curved Alexandrov spaces

We study Alexandrov spaces of nonnegative curvature whose boundaries consist of several strata of codimension 1. If the space is compact and the common intersection of all boundary strata is empty, then the space is a metric product. In particular, this is fulfilled if the compact space has dimension n and contains more than n+1 boundary strata. The splitting factors are in general non-flat.