ASTIS mathematical exposition

Positive log-concavity implies quasiconcavity

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

Statement

Let E be a real module and f:E→ℝ positive log-concave on s⊆E. Then f is quasiconcave on s: every superlevel set {x∈s:c≤f(x)}, for every real c, is convex.

\[\operatorname{LC}_s(f)\Longrightarrow\operatorname{Quasiconcave}_s(f)\equiv\forall c\in\mathbb R,\ \{x\in s:c\le f(x)\}\text{ convex}.\]

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.

{'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

1. Provide every superlevel-set instance

For an arbitrary threshold c, the already-proved convex_superlevel theorem gives exactly the convex set required by the definition of QuasiconcaveOn. Returning this family of proofs completes the assertion.

\[c\longmapsto[\{x\in s:c\le f(x)\}\text{ is convex}].\]

Corresponding Lean step

fun c => hf.convex_superlevel c

The identifiers refer to the assumptions or previously proved facts described in this step; the full original proof below supplies their exact context.

Lean statement · quasiconcaveOn

Braces name inputs Lean can infer, and bracketed classes state the ambient structures listed above. The assumptions before the final colon are inputs; the expression after it is the exact property this declaration establishes. fun c constructs a proof for every threshold. This declaration packages the preceding theorem in Mathlib's standard quasiconcavity predicate.

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

theorem LogConcaveOn.quasiconcaveOn {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) :
    QuasiconcaveOn ℝ s f

Exact module and namespace context

Lean proof · quasiconcaveOn

fun c constructs a proof for every threshold. This declaration packages the preceding theorem in Mathlib's standard quasiconcavity predicate.

theorem LogConcaveOn.quasiconcaveOn {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) :
    QuasiconcaveOn ℝ s f :=
  fun c => hf.convex_superlevel c

Exact module and namespace context

Scope and omitted-condition boundaries

Reuse wrapper around convex_superlevel, not an independent superlevel proof.
This documentation adds no Lean theorem, compilation evidence, source-equivalence verdict, or new source-fidelity certification.

Source and reuse

ASTIS parents called

AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.convex_superlevel

Mathlib API called (external library)

QuasiconcaveOn

Mathematical sources

Existing ASTIS declaration and exact proof — Directly read current local source; no Lean edit or fresh build.
Existing curated module card — Local API/source-boundary memory; not independent primary textbook verification.
QuasiconcaveOn — Exact inspected Mathlib definition or theorem used by this exposition.
Existing usage in Tests.Basic — Read-only source example; no test was run.

ASTIS prose is not a quotation or a source-equivalence certificate. Definitions and aliases are explained as constructions, not counted as new mathematical proofs.