Nonnegative real powers preserve positive log-concavity

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

Statement

Let E be a real module, f:E→ℝ positive log-concave on s⊆E, and p≥0 a real number. Then x↦f(x)^p, with real exponentiation, is positive log-concave on s. The exponent zero is allowed.

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.
• p : ℝ and hp : 0≤p.

• {'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

refine hscaled.congr ?_
intro x hx
simpa [smul_eq_mul] using (Real.log_rpow (hf.pos (x := x) hx) p).symm

theorem LogConcaveOn.rpow {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ} {p : ℝ}
    (hf : LogConcaveOn s f) (hp : 0 ≤ p) :
    LogConcaveOn s (fun x => (f x) ^ p)

theorem LogConcaveOn.rpow {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ} {p : ℝ}
    (hf : LogConcaveOn s f) (hp : 0 ≤ p) :
    LogConcaveOn s (fun x => (f x) ^ p) := by
  refine ⟨fun x hx => Real.rpow_pos_of_pos (hf.pos (x := x) hx) p, ?_⟩
  have hscaled : ConcaveOn ℝ s (fun x => p • Real.log (f x)) :=
    hf.concaveOn_log.smul hp
  refine hscaled.congr ?_
  intro x hx
  simpa [smul_eq_mul] using (Real.log_rpow (hf.pos (x := x) hx) p).symm

/-- Product-domain tensorization: the product of log-concave factors on
convex domains is log-concave on the Cartesian product. -/

Scope and omitted-condition boundaries

• No conclusion for negative p is asserted; negative scaling generally changes concavity to convexity.
• This documentation adds no Lean theorem, compilation evidence, source-equivalence verdict, or new source-fidelity certification.