def Talagrand.hypercube (n : ℕ) : Set (Fin n → ℝ)

The discrete hypercube $\{0, 1\}^n$, viewed as a subset of $\mathbb{R}^n$: vectors whose every coordinate is $0$ or $1$.

noncomputable def Talagrand.euclideanDist {n : ℕ} (x y : Fin n → ℝ) : ℝ

The standard Euclidean distance $\|x - y\|_2 = \sqrt{\sum_i (x_i - y_i)^{2}}$ on $\mathbb{R}^n$.

noncomputable def Talagrand.euclideanInfDist {n : ℕ} (x : Fin n → ℝ) (A : Set (Fin n → ℝ)) : ℝ

The Euclidean distance from a point $x \in \mathbb{R}^n$ to a set $A$: the infimum of $\|x - a\|_2$ over $a \in A$.

theorem Talagrand.talagrand_953 {n : ℕ} (μ : MeasureTheory.Measure (Fin n → ℝ)) [MeasureTheory.IsProbabilityMeasure μ] (hsupp : μ (hypercube n)ᶜ = 0) (hunif : ∀ x ∈ hypercube n, μ {x} = 1 / 2 ^ n) (A : Set (Fin n → ℝ)) (hA : Convex ℝ A) (t : ℝ) (ht : 0 ≤ t) : (μ A).toReal * (μ {x : Fin n → ℝ | t ≤ euclideanInfDist x A}).toReal ≤ Real.exp (-t ^ 2 / 4)

Talagrand's convex distance inequality (Theorem 9.5.3). If $x$ is uniformly distributed on the discrete hypercube $\{0, 1\}^n$ and $A \subseteq \mathbb{R}^n$ is any convex set, then $\mathbb{P}(x \in A) \cdot \mathbb{P}\!\big(\operatorname{dist}(x, A) \ge t\big) \le \exp(-t^{2}/4)$ for every $t \ge 0$.