logConcaveOn_const_mul_exp_neg_of_strongConvexOn

Plain-English statement

- Positive scalar normalization preserves the log-concavity of a Gibbs shape whose potential is strongly convex with nonnegative modulus.

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Lean statement

theorem logConcaveOn_const_mul_exp_neg_of_strongConvexOn
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    {s : Set E} {V : E → ℝ} {k c : ℝ}
    (hV : StrongConvexOn s k V) (hk : 0 ≤ k) (hc : 0 < c) :
    LogConcavity.LogConcaveOn s (fun x => c * Real.exp (-V x)) :=
  LogConcavity.logConcaveOn_const_mul_exp_neg_of_convexOn
    (convexOn_of_strongConvexOn_nonneg hV hk) hc

/-- A strongly convex function with a global minimizer has a centered quadratic
lower bound.

The constant `k / 4` is the midpoint consequence of Mathlib's
`StrongConvexOn` convention.  It is intentionally not the sharp `k / 2`
first-order/subgradient bound, because this leaf requires no differentiability
or limiting argument and is enough for Gibbs-tail integrability. -/

Open AutoSamplingTheory/TechnicalLemmas/Geometry/StrongConvexity.lean:42published source at 7bcd37294df1

Proof architecture

Chewi DENS/CONV root: keep positive scalar normalizers separate while preserving strong-convex Gibbs log-concavity

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