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ASTIS mathematical exposition

The log-concavity domain is convex

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

Statement

Let E be a real module and f:E→ℝ positive log-concave on s⊆E. Then s is convex.

\[\operatorname{LC}_s(f)\Longrightarrow\forall x,y\in s,\ a,b\ge0,\ a+b=1:\ ax+by\in s.\]

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, 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. Read the domain clause of log concavity

ConcaveOn(log f) contains convexity of s as its first field. Extract that field; it was required by the definition and is not newly inferred from a pointwise inequality on a nonconvex domain.

\[\operatorname{Concave}_s(\log f)=\bigl[s\text{ convex}\bigr]\land[\text{concavity inequality}].\]
Corresponding Lean step

hf.concaveOn_log.1

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 · convex_domain

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. The first .1 extracts the convex-domain component from ConcaveOn. The theorem is a convenient interface to stored information.

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

Exact module and namespace context

Lean proof · convex_domain

The first .1 extracts the convex-domain component from ConcaveOn. The theorem is a convenient interface to stored information.

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.convex_domain {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) :
    Convex ℝ s :=
  hf.concaveOn_log.1

/-- Superlevel sets of a positive log-concave function are convex within the
log-concavity domain. -/

Exact module and namespace context

Scope and omitted-condition boundaries

  • Do not describe this as proving convexity for an arbitrary support of a nonnegative density.
  • This documentation adds no Lean theorem, compilation evidence, source-equivalence verdict, or new source-fidelity certification.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

  • ConcaveOn

Mathematical sources

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