  We study a new construction of bodies from a given convex body in
$\mathbb{R}^{n}$ which are isomorphic to (weighted) floating bodies. We
establish several properties of this new construction, including its relation
to $p$-affine surface areas. We show that these bodies are related to Ulam's
long-standing floating body problem which asks whether Euclidean balls are the
only bodies that can float, without turning, in any orientation.
