ASTIS mathematical exposition

A positively rescaled convex-potential Gibbs shape is log-concave

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

Statement

Let E be a real module, s⊆E, V:E→ℝ convex on s, and c>0 a real constant. Then x↦c exp(−V(x)) is positive log-concave on s.

\[\operatorname{Convex}_s(V),\quad c>0\Longrightarrow\operatorname{LC}_s(ce^{-V}).\]

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, V : E → ℝ, hV : ConvexOn ℝ s V.
c : ℝ, hc : 0<c.

{'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. Apply the convex-potential result

The existing theorem proves positivity and log-concavity of exp(−V) on s from the supplied convexity.

\[\operatorname{Convex}_s(V)\Longrightarrow\operatorname{LC}_s(e^{-V}).\]

Corresponding Lean step

logConcaveOn_exp_neg_of_convexOn hV

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

2. Rescale by the positive constant

Apply positive-rescaling closure to c. Equivalently, the new logarithm is log c−V, a concave function obtained by a constant shift.

\[\log(ce^{-V})=\log c-V,\qquad ce^{-V}>0.\]

Corresponding Lean step

(logConcaveOn_exp_neg_of_convexOn hV).const_mul hc

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_mul_exp_neg_of_convexOn

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. This proof composes two existing results. c may later be chosen as a partition-function reciprocal, but the declaration assumes only c>0.

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_mul_exp_neg_of_convexOn {E : Type*}
    [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {V : E → ℝ} {c : ℝ}
    (hV : ConvexOn ℝ s V) (hc : 0 < c) :
    LogConcaveOn s (fun x => c * Real.exp (-V x))

Exact module and namespace context

Lean proof · logConcaveOn_const_mul_exp_neg_of_convexOn

This proof composes two existing results. c may later be chosen as a partition-function reciprocal, but the declaration assumes only c>0.

theorem logConcaveOn_const_mul_exp_neg_of_convexOn {E : Type*}
    [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {V : E → ℝ} {c : ℝ}
    (hV : ConvexOn ℝ s V) (hc : 0 < c) :
    LogConcaveOn s (fun x => c * Real.exp (-V x)) :=
  (logConcaveOn_exp_neg_of_convexOn hV).const_mul hc

/-- The absolute value is convex on the real line. -/

Exact module and namespace context

Scope and omitted-condition boundaries

Reuse/integration wrapper; no assertion that c normalizes an integral.
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)

No direct Mathlib call recorded; see the ASTIS parents.

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.
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.