Transport and tiling bounds for the affine plank problem on the triangle

Bang's affine plank conjecture asserts that planks covering a convex body have relative widths summing to at least $1$; it is open already in the plane. For a convex body K and directions given by affine maps uᵢ:K→[0,1] we introduce the transport defect DK(u) — the least D admitting a probability measure on K with all uᵢ-marginals ≤ D — so every covering by planks in these directions satisfies ∑ rw≥ 1/DK(u), and we prove a minimax duality for DK with exact certificates on both sides. On the triangle we determine exactly when the transport method certifies the sharp constant $1$: for three pairwise non-parallel directions, none parallel to a facet, a uniform-marginal witness measure exists ($D=1$) if and only if the mid-level lines concur; for every concurrent cyclic triple we exhibit the witness and the unique tight covering with ∑ rw=1 — a two-parameter family of rigid configurations extending the median case — and we classify all concurrent triples admitting a witness. We also determine, partly by an exact-rational computation, the one-plank-per-direction covering constant at one tilted non-concurrent triple (the three-plank bound). Calibrations: the uniform measure gives D≤ d on any d-dimensional body, and the d-simplex facets have $D=(d+1)/2$. No result resolves the conjecture for the triangle, let alone the plane. Mathematics Subject Classification: 52C17, 52A40, 52A10, 52C15