Jinyoung Park (박진영), A reformulation of Talagrand’s Discrete Convexity Conjecture

The “Convexity Conjecture” by Talagrand asks, roughly speaking, whether one can “create convexity” in a bounded number of steps regardless of the dimension of the ambient space. Talagrand also proposed a discrete version of this conjecture, calling it his “lifetime favorite problem” and offering a $1,000 prize for its solution. While the continuous version of the conjecture was recently proven by Hua, Song, and Tudose, the discrete analogue remains wide open. In this talk, we introduce a reformulation of the discrete convexity conjecture using the new notion of “k-thresholds,” an extension of the traditional definition of thresholds. Using this framework, we establish the conjecture for several special cases, focusing primarily on graph properties.