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Basic definition on convex set and convex function #44

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17 changes: 16 additions & 1 deletion Probability/Probability/Defs.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -209,7 +209,7 @@ theorem one_of_ind_bool_or_not : (𝕀∘B) + (𝕀∘(¬ᵣ B)) = (1 : FinRV n

variable {X Y: FinRV n ℚ}

theorem rv_le_abs : X ≤ abs ∘ X := by intro i; simp [le_abs_self (X i)]
theorem rv_le_abs(X : FinRV n ℚ) : X ≤ abs ∘ X := by intro i; simp [le_abs_self (X i)]

theorem rv_prod_sum_linear {Xs : Fin k → FinRV n ℚ} : ∑ i, Y * (Xs i) = Y * (∑ i, Xs i) :=
by ext ω
Expand Down Expand Up @@ -348,3 +348,18 @@ theorem exp_indi_eq_exp_indr : ∀i : Fin k, 𝔼[L =ᵢ i // P] = 𝔼[𝕀 ∘
theorem exp_monotone (h: X ≤ Y) : 𝔼[X // P] ≤ 𝔼[Y // P] := dotProduct_le_dotProduct_of_nonneg_left h P.nneg

end Expectation_properties

namespace Geometry

variable {E : Type} [AddCommGroup E] [Module ℚ E]

def convex (S : Set E) : Prop :=
∀ ⦃x y : E⦄ ⦃t : ℚ⦄, 0 ≤ t → t ≤ 1 → x ∈ S → y ∈ S → t • x + (1 - t) • y ∈ S

def convex_on (S : Set E) (f : E → ℚ) : Prop :=
convex S ∧ ∀ ⦃x y : E⦄ ⦃t : ℚ⦄, 0 ≤ t → t ≤ 1 → x ∈ S → y ∈ S →
f (t • x + (1 - t) • y) ≤ t * f x + (1 - t) * f y



end Geometry