Geometric and functional inequalities have long been foundational in analysis, shaping our understanding of numerous mathematical and physical phenomena. Recently, attention has turned to questions of stability: if a function is nearly optimal in a known inequality, can we quantify just how close it is to a genuine minimizer? In this talk, I will discuss how Optimal Transport offers a potent framework for tackling these stability questions—while also noting scenarios where OT may not be the ideal tool...
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