What was already known
Bang’s theorem addressed full coverings; Bezdek proved partial-covering cases for triangles and restricted plank families. The authors already had a Minkowski-subtraction framework.
AI involvement
The authors credit iterative ChatGPT discussions with the planar idea and concavity use within their existing approach. They developed the final proof.
The declaration specifically credits the planar idea emerging with ChatGPT, while locating it within the authors' existing method; this is central joint work, not model leadership.
Importance
Extends single-plank optimality from triangles to all planar convex bodies.
Limitations
Bakaev and Polyanskii explain the reduction through Minkowski subtraction. The paper also solves a centered-ball question in all dimensions, but its AI disclosure does not isolate that ball proof as model-origin. A completed check of the planar proof is not documented.
Technical verification
Theorem 12 and its proof are published in the v1 manuscript, but the reviewed source does not report a completed human check of that argument or a machine check. A proof's presence alone does not establish Formal verification.
Human understanding
Bakaev and Polyanskii explain the planar mechanism: cut the Minkowski difference into two pieces, translate them to the area outside a plank, and use the common reduction to prove optimality.
Correction and publication history
If correct, extends single-plank optimality from triangles to all planar convex bodies.
Read this published revision