Gaussian Inner-Product Second Moment

Proved

FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_inner_sq_readable

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

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This is the second-moment formula for the Gaussian law gaussianLaw.

Let $E$ be a finite-dimensional real inner-product space with Borel measurable structure, let $s>0$ , and let $v\in E$ . Then

\int_E\langle v,y\rangle^2\,d\gamma_s(y) =\frac{\|v\|^2}{2s}.

The result uses the Gaussian normalization encoded by gaussianLaw; completeness of $E$ is not required in the type. It is the raw second-moment identity from which the Gaussian variance formula follows.

Original Lean declaration header

Within namespace FunctionalInequalities.KLS.LogConcave.GaussianSmoothing. Shown in its original module context. Proof omitted.

theorem gaussianLaw_inner_sq {s : ℝ} (hs : 0 < s) (v : E) :
    (∫ y : E, ⟪v, y⟫ ^ 2 ∂(gaussianLaw s)) = ‖v‖ ^ 2 / (2 * s)

Module navigation

Module: KLS/LogConcave/Density/GaussianSmoothing/Moments.

Definition modules: KLS/LogConcave/Density/GaussianSmoothing/Basic.

Supporting theorems: gaussianLaw_exp_norm_integrable, gaussianLaw_laplace.

Previous: gaussianLaw_mean_zero · Next: smoothedLaw_mean.

Source excerpt, lines 166–167 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_LogConcave_Density_GaussianSmoothing_Moments

open scoped RealInnerProductSpace ContDiff
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [CompleteSpace E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
omit [CompleteSpace E]

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_inner_sq_readable {s : ℝ} (hs : 0 < s) (v : E) :
    (∫ y : E, ⟪v, y⟫ ^ 2 ∂(gaussianLaw s)) = ‖v‖ ^ 2 / (2 * s) := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/LogConcave/Density/GaussianSmoothing/Moments.lean#L164-L208 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.LogConcave.Density.GaussianSmoothing.Moments`, declaration `FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_inner_sq`; the repository is private at upload time).

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