Asymptotic Weighted Approximation of Convex Functions
Fernanda M. Baêta
TL;DR
The paper develops sharp asymptotics for approximating smooth convex functions with compact domain by circumscribed piecewise-affine convex functions, introducing a weighted error Δ_p(u, P_{(m)}^c, ω) that extends classical affine-surface notions to the function setting. Central to the results is the ζ-affine surface area functional Z_ζ(u) and Zador's constant δ_{p,n}, which together yield the exact limit lim_{m→∞} m^{2p/n} Δ_p(u, P_{(m)}^c, ω) = (δ_{p,n}/2^p) ( ∫_{dom u} (det D^2 u)^{p/(n+2p)} ω(x,u(x))^{n/(n+2p)} dx )^{(n+2p)/n} for C^2_+ functions, with unweighted and p=1 cases recovering Gruber-type results. The work also develops a dual functional-analytic perspective via Legendre transform and Monge–Ampère measure, and extends the framework to include the weighted functional affine surface area, connecting to recent developments in convex-analytic geometry. These results provide sharp, dimension-dependent rates for functional polytopal approximation and establish deep links between Hessian determinants, affine geometry, and duality in convex analysis.
Abstract
Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to infinity. These asymptotic expressions are formulated in terms of a functional that extends the notion of affine surface area to the functional setting.