  We investigate a natural analog to Lutwak's $p$-affine surface area in
$d$-dimensional spherical, hyperbolic and de Sitter space. In particular, we
show that these curvature measures appear naturally as the volume derivative of
floating bodies of non-Euclidean convex bodies conjugated by duality, such as
spherical, hyperbolic and de Sitter convex bodies.
  We provide a unifying framework by establishing a real-analytic version of
this relation controlled by the constant curvature of the $d$-dimensional real
space form. These new curvature measures relate in two distinctly different
ways to curvature measures on Euclidean space, one of which is Lutwak's
centro-affine invariant $p$-affine surface area, and the other is related to a
rigid-motion invariant curvature measure that appears naturally as the volume
derivative of Schneider's mean-width separation body.
