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arXiv · 2607.24388

Sharpness and Stability of the Alexandrov-Bakelman-Pucci Estimate: a Convex Geometric Approach

Antonio Porricelli
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Abstract

Based on a 1981 work by G. Talenti, which established a comparison principle for smooth solutions of the Monge-Amp\`ere equation alongside related two-dimensional a priori estimates, this paper explores the connection to the Blaschke-Santal\'o inequality and its reverse form by K. Mahler (1939). More specifically, in this work, a quantitative version of one of Talenti's a priori estimates is proved. We exploit the interplay between Talenti's analytical framework and Mahler/Blaschke-Santal\'o 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.

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Antonio Porricelli. 2026-07-27. Sharpness and Stability of the Alexandrov-Bakelman-Pucci Estimate: a Convex Geometric Approach. https://arxiv.org/abs/2607.24388

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