arXiv:2610.31546v1 Announce Type: new Abstract: We determine when approximation by convex polynomials on expanding intervals converges to best approximation on the real line by convex entire functions of exponential type at most $\sigma>0$. For intervals of radius $(n-\gamma_n)/\sigma$, where $0\le\gamma_nSource:https://arxiv.org/abs/2610.31546