A uniform upper bound $d_p$ for the norm Hessian at a near-coordinate vector

Proved

HlawkaSchatten.DiagonalConstruction.normHessian_upper

by savarin · Sep 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

convexityhessianhlawka-schattenlp-norm

For $v,h\in\mathbb R^3$ , define

\mathrm{powerSum}_p(v):=\sum_i|v_i|^p,\qquad \mathrm{radialCoefficient}_p(v,h):=\frac{\sum_i|v_i|^{p-2}v_ih_i}{\mathrm{powerSum}_p(v)},

\mathrm{normHessian}_p(v,h):=(p-1)\,\mathrm{powerSum}_p(v)^{1/p-1}\sum_i|v_i|^{p-2}\bigl(h_i-\mathrm{radialCoefficient}_p(v,h)\,v_i\bigr)^2,

and for $p>2$ define

d_p := \frac{2(p-1)(19/50)^{p-2}}{(81/50)^{p-1}}.

For every $p>2$ , every $v,h\in\mathbb R^3$ , and every coordinate $k\in\{0,1,2\}$ such that $v_k$ is large, $81/50\le|v_k|$ , while every other coordinate is small, $|v_i|\le19/50$ for $i\ne k$ ,

\mathrm{normHessian}_p(v,h) \;\le\; d_p\,\lVert h\rVert_2^2,\qquad \lVert h\rVert_2^2=\textstyle\sum_ih_i^2.

This is a uniform upper curvature bound for the finite coordinate $p$ -norm's Hessian, at a vector with one dominant, nearly saturated coordinate and two small ones — the shape taken, for instance, by a pairwise column sum such as $X_0+X_1$ of a triple confined to a box around the cyclic sign pattern.

Formalization Note No hypothesis that $v\ne0$ is needed: $81/50\le|v_k|$ already forces $v_k\ne0$ , hence $\mathrm{powerSum}_p(v)>0$ , so $\mathrm{normHessian}_p(v,h)$ is well-defined without a division-by-zero or zero-to-a-negative-power issue, even though the other two coordinates of $v$ may vanish.

Preamble

import Definitions.Def_HlawkaSchatten_DiagonalConstruction_BoxGeometry
import Definitions.Def_HlawkaSchatten_DiagonalConstruction_HessianBounds
import Definitions.Def_HlawkaSchatten_DiagonalConstruction_NormHessian
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Convex.Deriv
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.NormPow
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Real.Basic
import Mathlib.Data.Sign.Basic
import Mathlib.Tactic.Abel
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Topology.Instances.Sign
import Mathlib.Topology.Order.Compact

/-
Copyright (c) 2026 Ezzeri Esa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ezzeri Esa
-/

/-! # Uniform lower and upper bounds for the norm Hessian -/

open HlawkaSchatten.DiagonalConstruction

Formal statement

theorem HlawkaSchatten.DiagonalConstruction.normHessian_upper {p : ℝ} (hp : 2 < p) (v h : Fin 3 → ℝ) (k : Fin 3)
    (hk : 81 / 50 ≤ |v k|) (hi : ∀ i, i ≠ k → |v i| ≤ 19 / 50) :
    normHessian p v h ≤ upperHessianCoefficient p * euclideanSq h := by sorry

Source

https://github.com/savarin/hlawka-schatten/blob/79aa498bfcf7b22bd91d771fb32ec278e2d4704b/HlawkaSchatten/DiagonalConstruction/HessianBounds.lean#L102-L145

Human review

Endorsed by Shuze Chen · Sep 29, 2026
Confirmed by the moderator at approval.
Endorsed by savarin · Sep 29, 2026
Confirmed by the mission captain (proposal self-audit).
Endorsed by marwahaha · Sep 30, 2026

View graph

A uniform upper bound dpd_pdp​ for the norm Hessian at a near-coordinate vector

A uniform upper bound $d_p$ for the norm Hessian at a near-coordinate vector