Nonconvex Chebyshev sets in infinite-dimensional Hilbert spaces

Determine whether an infinite-dimensional Hilbert space admits a nonconvex Chebyshev set.

Background

The paper discusses proximinal and Chebyshev sets in connection with disjoint tilings and Voronoi cells. A Chebyshev set is a subset of a metric space for which every point has a unique nearest point in the set.

The authors identify the existence of nonconvex Chebyshev sets in infinite-dimensional Hilbert spaces as a well-known unresolved problem in approximation theory. This problem is presented as related background rather than as a question resolved by the tiling results proved in the paper.

References

It could be added that one of the well-known open problems in approximation theory is whether an infinite-dimensional Hilbert space admits a nonconvex Chebyshev set ().

— On disjoint tilings of $\ell_1(κ)$ by star-shaped bodies  (2609.35180 - Koszmider et al., 28 Sep 2026) in Section 1, Introduction