For a connected -dimensional compact smooth hypersurface M without boundary embedded in a (n + 1)-dimensional Euclidean space, a classical result of A. D.
Alexsandrov shows that it must be a sphere if it has constant mean curvature. Nirenberg and I studied a one-directional analog of this result: if every pair of points (x', a), (x', b) € M with a < 6 has ordered mean curvature H(x', b) < H(x', a), then M is symmetric about some hyperplane *n+1 = c under some additional conditions. A conjecture of theirs was recently proved in a joint work with Xukai Yan and Yao Yao. These works and some open problems will be presented in this talk.