Unpack the definition of positive log-concavity

AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_iff · theorem · Teaching coverage

Statement

For any real module E, set s⊆E and function f:E→ℝ, positive log-concavity on s is equivalent to strict positivity of f on s together with concavity of its logarithm on s.

All objects and hypotheses

• E : Type* with [AddCommMonoid E] and [Module ℝ E]. These are the exact algebraic ambient structures; no topology, norm, measure or finite-dimensionality is assumed.
• s : Set E; f : E → ℝ; no additional hypothesis.

• {'term': 'Positive log-concavity', 'text': 'LC_s(f) means strict positivity of f at every point of s and concavity of log f on the convex domain s. No zero-valued points are included in this convention.', 'formula': '\\operatorname{LC}_s(f)\\iff(\\forall x\\in s,\\ f(x)>0)\\land\\operatorname{Concave}_s(\\log f).'}
• {'term': 'Convex combinations and Jensen inequalities', 'text': 'Throughout, x,y lie in the stated domain; λ,θ are nonnegative real weights with λ+θ=1. The Lean code often names these weights a,b, independently of the a,b coefficients in specialized potentials.', 'formula': 'z=\\lambda x+\\theta y,\\quad \\lambda,\\theta\\ge0,\\quad\\lambda+\\theta=1;\\qquad V(z)\\le\\lambda V(x)+\\theta V(y)\\text{ for convex }V.'}
• {'term': 'Geometry, not probability normalization', 'text': 'The module proves shapes and convexity properties of real-valued functions. It has no reference measure in its declarations. In particular, names containing normalized_density do not themselves prove normalization, and the quadratic prefactor is not certified as the integral of an arbitrary norm-based shape.', 'formula': '\\operatorname{LC}(Z^{-1}e^{-V})\\quad\\text{does not assert}\\quad Z=\\int e^{-V}\\,d\\mu\\quad\\text{or}\\quad \\int Z^{-1}e^{-V}\\,d\\mu=1.'}

Mathematical proof

theorem logConcaveOn_iff {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ} :
    LogConcaveOn s f ↔
      (∀ x ∈ s, 0 < f x) ∧ ConcaveOn ℝ s (fun x => Real.log (f x))

theorem logConcaveOn_iff {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ} :
    LogConcaveOn s f ↔
      (∀ x ∈ s, 0 < f x) ∧ ConcaveOn ℝ s (fun x => Real.log (f x)) :=
  Iff.rfl

Scope and omitted-condition boundaries

• Definition-level interface only; not a new mathematical characterization by geometric means or densities.
• This documentation adds no Lean theorem, compilation evidence, source-equivalence verdict, or new source-fidelity certification.