A positive log-concave function is positive at each domain point

AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.pos · theorem · Teaching coverage

Statement

Let E be a real module, f:E→ℝ positive log-concave on s⊆E, and x∈s. Then f(x)>0.

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, hf : LogConcaveOn s f.
• x : E with hx : x∈s.

• {'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.pos {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) {x : E} (hx : x ∈ s) :
    0 < f x

theorem LogConcaveOn.pos {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) {x : E} (hx : x ∈ s) :
    0 < f x :=
  hf.1 x hx

Scope and omitted-condition boundaries

• No assertion about f outside s; positivity is extracted from the chosen definition, not deduced from concavity of log alone.
• This documentation adds no Lean theorem, compilation evidence, source-equivalence verdict, or new source-fidelity certification.