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 Ulams long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.
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