Explicit shapes of N-dimensional neutral floating bodies

Determine explicit geometries of bodies in N-dimensional Euclidean space that float in equilibrium in all orientations (neutral floating bodies) in those cases where existence has been established, by constructing or characterizing concrete shapes realizing the proven existence results.

Background

Ulam’s floating body problem asks whether there exist homogeneous solids that float in equilibrium in all orientations. In two dimensions, non-circular solutions exist and are described by Zindler curves. Recent advances address higher-dimensional settings, where certain existence results are known for bodies that float in all orientations.

Despite these existence results in higher dimensions, explicit constructions or characterizations of the corresponding shapes are not currently known. The paper highlights this gap, noting that while some N-dimensional cases have established existence, concrete shapes realizing these solutions have not been identified.

References

Significant progress has also been made recently in the general N-dimensional case, where the existence of bodies floating in all orientations has been established in some cases, although their explicit shapes remain unknown [11-13].

— A floating body with no preferred orientation: an experimental realization  (2604.01692 - Pontiggia et al., 2 Apr 2026) in Introduction