Quantization Coefficients for Uniform Distributions on the Boundaries of Regular Polygons

The paper proves lim n^2 V_n = (1/3) m^2 sin^2(π/m) by (i) solving the 1D segment problem (Theorem 2.1), (ii) deriving an exact formula for V_n when n = mk (Proposition 2.3), and (iii) sandwiching general n via (mℓ(n))^2 V_{m(ℓ(n)+1)} < n^2 V_n < (m(ℓ(n)+1))^2 V_{mℓ(n)}, yielding the limit (Theorem 2.4) . The candidate’s upper bound is fine, but the lower-bound argument has a critical gap: it asserts one can partition the codebook into m pairwise disjoint nonempty sub-codebooks (one per trimmed side) and then apply Jensen with ∑ k_i ≤ n. In general this disjoint, all-nonempty assignment need not exist (a single center’s Voronoi region can meet multiple trimmed sides), so the step that forces the extra factor m^2 is not justified as written. The limit value is still correct, but the candidate’s proof of the lower bound is logically incomplete.