Convergence and finite length of the Brascamp–Lieb scaling flow

We give a proof of full convergence of the prescribed Brascamp–Lieb scaling flow for every feasible real datum with positive real weights, answering Question 2.10 of Bez, Gauvan, and Tsuji. The limit is geometric, and the sequence of complete scaling iterations has finite length in the product Frobenius norm. Stacking the matrices with square-root weights identifies the two normalizations with nearest-point projections onto a product of scaled row-Stiefel manifolds and a column-Stiefel manifold. The result then follows from the asymptotic normalization theorem of Bez, Gauvan, and Tsuji and the established local alternating-projection theory of Noll and Rondepierre. We also give a direct verification of the descent conditions of Attouch, Bolte, and Svaiter. The one-dimensional-output recurrence agrees with the prescribed-norm frame algorithm studied by Zhu and Li. An explicit feasible example has infinite interleaved half-step length and error of order \(k^{-1/2}\), clarifying the scope of the complete-iteration conclusion.