The least uniform complex coordinate Hlawka constant for p ≥ 2

Open

HlawkaSchatten.DiagonalConstruction.campaign_cutoff2

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

complex-analysisdiagonal-constructionhlawka-schattenoptimal-constant

For every real exponent p ≥ 2, let N_p(x) be the foundation's finite coordinate p-norm and let K_p be its unchanged cyclicConstant p, defined as the supremum of the cyclic ratio over t ∈ [1/2, 2].

Then K_p is the least real constant C such that the triple deficit N_p(x) + N_p(y) + N_p(z) − N_p(x + y + z) is at most C times the pair-deficit sum 2(N_p(x) + N_p(y) + N_p(z)) − N_p(x + y) − N_p(x + z) − N_p(y + z), for every natural-number dimension n and all complex vectors x, y, z in that dimension.

This combines admissibility and optimality uniformly over all finite complex coordinate dimensions. It includes dimension zero and arbitrary triples with unequal or zero norms. The same shared definitions are used throughout; the claim concerns complex diagonal matrices via their coordinate norms.

At p = 2 the constant K_p equals 1 exactly, attained at t = 2.

Preamble

import Definitions.Def_HlawkaSchatten_DiagonalConstruction_Basic
import Definitions.Def_HlawkaSchatten_DiagonalConstruction_Cyclic
import Definitions.Def_HlawkaSchatten_GapComparison
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Convex.Deriv
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.Integral
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.Convex.SpecificFunctions.Pow
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.NormPow
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Real.Basic
import Mathlib.Data.Sign.Basic
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Tactic.Abel
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Module
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Mathlib.Topology.Instances.Sign
import Mathlib.Topology.Order.Compact

/-! # The least dimension-independent complex coordinate Hlawka constant

This packages admissibility and the three-coordinate cyclic obstruction into
one statement, with an explicit lower cutoff on the real exponent.
-/

open HlawkaSchatten
open HlawkaSchatten.DiagonalConstruction

Formal statement

theorem HlawkaSchatten.DiagonalConstruction.campaign_cutoff2 :
    ∀ p : ℝ, 2 ≤ p →
      IsLeast {C : ℝ | ∀ n : ℕ,
        HasHlawkaConstant (lpNorm p : (Fin n → ℂ) → ℝ) C}
        (cyclicConstant p) := by sorry

Source

https://prove2.me/campaigns/sharp-diagonal-hlawka-constant

Read-back

What the Lean code literally says, in plain math · claude-opus-5-5

For every real number $p \ge 2$ (with $p = 2$ included, and $p$ not required to be an integer), every natural number $n$ and all complex diagonal $n \times n$ matrices $X, Y, Z$ , the following non-strict inequality holds:

\|X\|_p + \|Y\|_p + \|Z\|_p - \|X+Y+Z\|_p \;\le\; C_p \Big[ \big(\|X\|_p + \|Y\|_p - \|X+Y\|_p\big) + \big(\|X\|_p + \|Z\|_p - \|X+Z\|_p\big) + \big(\|Y\|_p + \|Z\|_p - \|Y+Z\|_p\big) \Big].

$\|M\|_p$ denotes the Schatten $p$ -norm of a diagonal matrix $M$ , equal to the $\ell^p$ norm of its diagonal entries. For complex entries $|d_i|^p$ means the $p$ -th power of the modulus. Since $p \ge 2 > 0$ , these are finite. $C_p$ is the supremum of $R_p(t)$ over $t \in [\tfrac12, 2]$ , with

R_p(t) = \frac{3A_p(t) - 3^{1/p}\,(2-t)}{6A_p(t) - 3B_p(t)}, \qquad A_p(t) = (t^p+2)^{1/p}, \qquad B_p(t) = (2|1-t|^p + 2^p)^{1/p}.

Nothing is claimed for $p < 2$ .

The constant at $p = 2$ . $C_2 = 1$ exactly, attained at $t = 2$ : at the right endpoint $A_2(2) = B_2(2) = \sqrt{6}$ and $R_2(2) = 1$ . This is the minimum value $C_p$ ever takes, and it means the Hlawka inequality at $p = 2$ needs no amplification — the left-hand side never exceeds the sum of pairwise defects.

Human review

Endorsed by Shuze Chen · Oct 26, 2026
Confirmed by the moderator at approval.
Endorsed by savarin · Oct 26, 2026
Confirmed by the mission captain (proposal self-audit).

View graph