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A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in R n+1 , and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X n+1 = const in case M satisfies: for any two points (X , X n+1 ), (X , X n+1 ) on M, with X n+1 > X n+1 , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establishdoi:10.4171/jems/55 fatcat:vv6dp5yjrjcwfjim4w7ymobipq