Smooth Relaxation Bound for Admissible Local-Slope Tests

Proved

FunctionalInequalities.Energy.SmoothGradient.relaxation_le_localSlope_of_admissible_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This statement extends the local-slope recovery bound from globally Lipschitz tests to all admissible tests.

Let $V$ be a finite-dimensional real inner-product space with complete Borel measurable and second-countable structure, and let $\nu$ be a probability measure on $V$ . Let $g:V\to\mathbb R$ satisfy LocalSlope.Admissible ν g, which supplies the admissibility, local Lipschitz regularity, $L^2(\nu)$ membership, measurability, and finite local-slope energy required by the definition.

Then

\operatorname{relaxation}(\nu)([g]_{L^2(\nu)}) \leq \operatorname{LocalSlope.energy}(\nu,g).

The energies are $[0,\infty]$ -valued, and the admissibility hypothesis in particular ensures the relevant $L^2$ representative is available. This is the general density estimate for the local-slope energy.

Original Lean declaration header

Within namespace FunctionalInequalities.Energy.SmoothGradient. Shown in its original module context. Proof omitted.

theorem relaxation_le_localSlope_of_admissible (ν : ProbabilityMeasure V) {g : V → ℝ}
    (hg : LocalSlope.Admissible ν g) :
    relaxation ν (hg.memLp_two.toLp g) ≤ LocalSlope.energy ν g

Module navigation

Module: Energy/Euclidean/Slope/Density.

Definition modules: Energy/LocalSlope/Basic, Energy/Form/Functional, Energy/SmoothGradient/Basic.

Selected supporting theorems: apply_mul_le, lowerSemicontinuous_secant, lowerSemicontinuous_sequentialRelaxation, relaxation_le_localSlope_of_lipschitzWith, exists_lipschitzWith_of_eq_zero_outside_ball.

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Source excerpt, lines 1016–1018 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_Energy_Euclidean_Slope_Density

open MeasureTheory Filter Topology Set
universe u
variable {V : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V]
  [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] [FiniteDimensional ℝ V]

Formal statement


theorem FunctionalInequalities.Energy.SmoothGradient.relaxation_le_localSlope_of_admissible_readable (ν : ProbabilityMeasure V) {g : V → ℝ}
    (hg : LocalSlope.Admissible ν g) :
    relaxation ν (hg.memLp_two.toLp g) ≤ LocalSlope.energy ν g := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/Energy/Euclidean/Slope/Density.lean#L1013-L1187 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.Energy.Euclidean.Slope.Density`, declaration `FunctionalInequalities.Energy.SmoothGradient.relaxation_le_localSlope_of_admissible`; the repository is private at upload time).

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