Shortest Minkowski Billiard Trajectories on Convex Bodies

The paper’s Theorem 1.5 is established by a clean reduction-and-compactness strategy: (i) first show U is a linear subspace with dim U ≤ m−1 (Proposition 3.7), (ii) then upgrade to dim U = m−1 via a strict shortening argument using Proposition 3.10, the strict subadditivity of the support function (Proposition 2.3(i)), and the existence/translation result (Theorem 3.12), and (iii) finally prove dim(N_K(q_j) ∩ U) = 1 by a boundedness/decomposition argument that crucially uses smoothness of T (Proposition 3.8 and the boundedness of the intersection (44)) to contradict minimality if the intersection had higher dimension. This proof is explicit in the text and uses only ingredients available under the stated hypotheses . By contrast, the candidate solution relies on a variational deformation of the dual polygon p^t defined by perturbing the coefficients in p_{j+1}^t − p_j^t = −(μ_{j+1} + t α_{j+1}) n_K(q_{j+1}). It does not ensure the perturbed vertices p_j^t remain on ∂T, nor does it justify a unique, well-defined choice of primal edges e_j(t) ∈ N_T(p_j^t) (the selection is only unique up to positive scaling on a ray), and it invokes a second-derivative/Hessian argument for h_T that is delicate in view of its positive homogeneity. The step attempting to force dim(N_K(q_j) ∩ U) = 1 via a local split of a dual step similarly assumes a split with p̃ ∈ ∂T and appeals to “strict subadditivity” in a way that does not track the necessary normal–support correspondence on T. These gaps mean the model’s proof outline is not correct as written, while the paper’s argument is sound and complete under the stated hypotheses (strict convexity and smoothness of T) .