English
For a convex set s in a linearly ordered space, if f satisfies a ≤-type inequality for x<y with positive a,b, then concavity holds on s.
Русский
Для выпуклого множества s в линейно упорядоченном пространстве, если f удовлетворяет неравенству для x<y с положительными a,b, то неравенство выпуклости выполняется на s.
LaTeX
$$$\text{If } f \text{ is concave on } s, \; f\text{ is concave on } s$ under linear order assumptions.$$
Lean4
/-- For a function on a convex set in a linearly ordered space (where the order and the algebraic
structures aren't necessarily compatible), in order to prove that it is strictly convex, it suffices
to verify the inequality `f (a • x + b • y) < a • f x + b • f y` for `x < y` and positive `a`, `b`.
The main use case is `E = 𝕜` however one can apply it, e.g., to `𝕜^n` with lexicographic order. -/
theorem strictConvexOn_of_lt (hs : Convex 𝕜 s)
(hf :
∀ ⦃x⦄,
x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) < a • f x + b • f y) :
StrictConvexOn 𝕜 s f := by
refine ⟨hs, fun x hx y hy hxy a b ha hb hab => ?_⟩
wlog h : x < y
· rw [add_comm (a • x), add_comm (a • f x)]
rw [add_comm] at hab
exact this hs hf y hy x hx hxy.symm b a hb ha hab (hxy.lt_or_gt.resolve_left h)
exact hf hx hy h ha hb hab
