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ASTIS mathematical exposition

Positive constants are log-concave on convex domains

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

Statement

Let E be a real module, s⊆E a convex set, and c>0 a real number. The constant function x↦c is positive log-concave on s.

\[c>0,\quad s\text{ convex}\Longrightarrow\operatorname{LC}_s(x\mapsto c).\]

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; c : ℝ; hc : 0<c; hs : Convex ℝ 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

1. Check the two constant-function clauses

Every function value is the supplied positive c. Its logarithm is constant log c, whose concavity inequality is equality because the weights sum to one; the domain is convex by hs.

\[c>0,\qquad a\log c+b\log c=\log c\quad(a+b=1).\]
Corresponding Lean step

⟨fun _ _ => hc, concaveOn_const (Real.log c) hs⟩

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 · logConcaveOn_const

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. Both the positive value and convex domain are explicit inputs. A constant function need not integrate to a finite mass on an infinite-volume domain.

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_const {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {c : ℝ} (hc : 0 < c) (hs : Convex ℝ s) :
    LogConcaveOn s (fun _ : E => c)

Exact module and namespace context

Lean proof · logConcaveOn_const

Both the positive value and convex domain are explicit inputs. A constant function need not integrate to a finite mass on an infinite-volume domain.

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_const {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {c : ℝ} (hc : 0 < c) (hs : Convex ℝ s) :
    LogConcaveOn s (fun _ : E => c) :=
  ⟨fun _ _ => hc, concaveOn_const (Real.log c) hs⟩

/-- If `V` is convex, then the unnormalized Gibbs shape `exp (-V)` is
log-concave. -/

Exact module and namespace context

Scope and omitted-condition boundaries

  • No measure or normalizing claim.
  • 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_const

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.