Gaussian Linear Variance

Proved

FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_variance_readable

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

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This is the variance formula for linear observables of gaussianLaw.

Let $E$ be a complete finite-dimensional real inner-product space with Borel measurable structure, let $s>0$ , and let $v\in E$ . For the linear observable $y\mapsto\langle v,y\rangle$ and $\gamma_s=\texttt{gaussianLaw }s$ ,

\operatorname{Var}_{\gamma_s}\langle v,\cdot\rangle =\frac{\|v\|^2}{2s}.

The declaration includes the required completeness, finite-dimensionality, and Borel assumptions. It identifies the variance parameter used by the Gaussian smoothing construction.

Original Lean declaration header

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

theorem gaussianLaw_variance {s : ℝ} (hs : 0 < s) (v : E) :
    variance (fun y : E => ⟪v, y⟫) (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_inner_sq, gaussianLaw_mean_zero.

Previous: smoothedLaw_mean · Next: smoothedLaw_variance.

Source excerpt, lines 238–239 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_LogConcave_Density_GaussianSmoothing_Moments

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

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_variance_readable {s : ℝ} (hs : 0 < s) (v : E) :
    variance (fun y : E => ⟪v, y⟫) (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#L237-L253 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.LogConcave.Density.GaussianSmoothing.Moments`, declaration `FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_variance`; the repository is private at upload time).

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