For a connected n-dimensional compact smooth hypersurface M without boundary embedded in \({\mathbb {R}}^{n+1}\), a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a one-directional analog of this result: if every pair of points \((x',a), (x',b)\in M\) with \(a<b\) has ordered mean curvature \(H(x',b)\le H(x',a)\), then M is symmetric about some hyperplane \(x_{n+1}=c\) under some additional conditions. Their proof was done by the moving plane method and some variations of the Hopf Lemma. We obtain the symmetry of M under some weaker assumptions using a variational argument, giving a positive answer to the conjecture in [13].