An analogue of the Blaschke-Santaló inequality for billiard dynamics

The paper defines the billiard product β(K) = inf_z α(K) α(K^z) and proves the sharp bound β(K) ≤ 16 in all dimensions, with β(B) = 16 for balls, and in the plane it identifies the equality case with constant width bodies via the identity β(K) = 8 α(K)/diam(K) and a pair of elementary lemmas relating segments and slabs (Lemma 3.2) and the planar lower/upper bounds on α(K^z) (eq. (4.1)) . By contrast, the model’s proof hinges on the blanket claim α(K) = 2·w_min(K) for every convex body, which is false in general (the paper only uses α(K) ≤ 2·w(K) and α(K) ≤ 2·diam(K) and shows equality only in special cases) . The model also misapplies the Bezdek–Bezdek non-translatability criterion to arbitrary billiard trajectories, whereas that criterion characterizes minimizers in a specific variational set (Eq. (2.1)) rather than all trajectories . Although the model arrives at the correct numerical bound 16, its core step is invalid, so the argument is incorrect.