  We investigate weighted floating bodies of polytopes. We show that the
weighted volume depends on the complete flags of the polytope. This connection
is obtained by introducing flag simplices, which translate between the metric
and combinatorial structure.
  Our results are applied in spherical and hyperbolic space. This leads to new
asymptotic results for polytopes in these spaces. We also provide explicit
examples of spherical and hyperbolic convex bodies whose floating bodies behave
completely different from any convex body in Euclidean space.
