Talagrand's discrete convexity conjecture

Theorem 1.1: with $k=2^{75}$, for every $N\ge1$, $p\in(0,1)$ and $\mathcal D\subseteq2^{[N]}$, $\mu_p(\mathcal D)\ge1-1/k$ implies $E_k(\mathcal D)$ is $p$-small. Consequences: a containment cover for large values of positive selector processes (recovering Park-Pham's theorem), and with Li's fractional result and the companion rounding theorem, $E_2(\mathcal D)$ is $(p/C)$-small when $\mu_p(\mathcal D)>1/2$. Earlier, Hua-Song-Tudose derived a discrete statement with three unions at density $p^C$ and Park one with two unions at a $\log\log(1/p)$ loss. The constant $k$ is far from optimal and the Gaussian convexity question is a different problem.

Theorem 1.1: with $k=2^{75}$, for every $N\ge1$, $p\in(0,1)$ and $\mathcal D\subseteq2^{[N]}$, $\mu_p(\mathcal D)\ge1-1/k$ implies $E_k(\mathcal D)$ is $p$-small. Consequences: a containment cover for large values of positive selector processes (recovering Park-Pham's theorem), and with Li's fractional result and the companion rounding theorem, $E_2(\mathcal D)$ is $(p/C)$-small when $\mu_p(\mathcal D)>1/2$. Earlier, Hua-Song-Tudose derived a discrete statement with three unions at density $p^C$ and Park one with two unions at a $\log\log(1/p)$ loss. The constant $k$ is far from optimal and the Gaussian convexity question is a different problem.