Multivariate Lebesgue bounds for R-Leja lower grids

We establish an algebraic upper bound for the Lebesgue constant of multivariate polynomial interpolation on R-Leja lower grids. For each fixed dimension d≥2 and every nonempty finite lower set A⊂ N₀ᵈ, let PA denote the corresponding grid with $N=|A|$ nodes. We prove Λ(PA)≤ CdNβd, where qd=d/(d-1) and βd=1+qd⁻¹log₂(1+2qd). The exponent satisfies β₂=2.160964… and increases to log₂ 6=2.584962… as d→∞, improving both the cubic estimate and the exponent $3-1/d$ obtained from quadratic prefix bounds. Our proof exploits the binary structure of known univariate estimates to control sums of powers of Lebesgue constants over interpolation prefixes. A weighted Hölder inequality and a disjoint-box packing argument then transfer these prefix-moment bounds to arbitrary lower sets. The resulting estimate is uniform over all lower-set shapes and choices of disk Leja maximizers, with constants depending only on d.