Sharpness and planar stability of the Alexandrov-Bakelman-Pucci estimate: a convex geometric approach

Abstract Based on a 1981 work by G. Talenti, which established a comparison principle for smooth solutions of the Monge-Ampère equation alongside related two-dimensional a priori estimates, this paper explores the connection to the Blaschke-Santaló inequality and its reverse form by [17] . More specifically, in this work, a quantitative version of one of Talenti’s a priori estimates is proved. I exploit the interplay between Talenti’s analytical framework and Mahler/Blaschke-Santaló inequalities via convex geometry, demonstrating how this link can be utilized to gain sharp insights into the Alexandrov-Bakelman-Pucci (ABP) maximum principle for elliptic PDEs, deducing a stability result in dimension two involving such estimate.