  We use the weighted Hsiung-Minkowski integral formulas and Brendle's
inequality to show new rigidity results. First, we prove Alexandrov type
results for closed embedded hypersurfaces with radially symmetric higher order
mean curvature in a large class of Riemannian warped product manifolds,
including the Schwarzschild and Reissner-Nordstr\"{o}m spaces, where the
Alexandrov reflection principle is not available. Second, we prove that, in
Euclidean space, the only closed immersed self-expanding solitons to the
weighted generalized inverse curvature flow of codimension one are round
hyperspheres.
