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We study a new construction of bodies from a given convex body in Rn 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.

Given a Borel measure µ on R n , we define a convex set bywhere the union is taken over all µ-measurable functions f : R n → [0, 1] with R n f dµ = 1. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we provide a lower bound for certain average norms of centroid bodies of non-degenerate probability measures. (B. A. Slomka).

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