---
title: "An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound"
authors: ["Shuai Zeng"]
year: 2026
venue: "arXiv:math.MG"
arxiv: "2609.01284"
doi: null
source: "https://arxiv.org/abs/2609.01284"
topics: ["curiosity", "maths-foundations"]
seed_rank: 1547
seed_batch: "curiosity-2026-09-03"
reviewed: "2026-09-03"
pool: "maths-foundations"
relevance_score: 10
lineage: discrete-geometry
cites:
  - title: "An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound"
    url: "https://arxiv.org/abs/2609.01284"
    year: 2026
    arxiv: "2609.01284"
    doi: null
see: []
---

# An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound

## One-sentence takeaway

Lebesgue's 1914 universal covering constant is the limit of an exact Reuleaux-type finite-arc hierarchy Λ_M, with a certified lower bound 0.834.

## Why it's lovely

Why you might love this: a century-old "smallest convex set that covers every diameter-1 set" problem gets an exact variational hierarchy instead of ad-hoc covers. Each Λ_M is a continuous finite-dimensional problem; the covering constant is their monotone limit, and the paper ships a machine-checked-style certified 0.834 lower bound. Classical convex geometry with modern certification teeth.

## Key ideas

- Lebesgue 1914: smallest-area planar convex set containing a congruent copy of every set of diameter ≤ 1.
- Exact Reuleaux-type variational hierarchy: monotone finite-arc values Λ_M with a_Leb = lim Λ_M.
- Each level is a continuous finite-dimensional optimization problem.
- Certified lower bound 0.834 on a_Leb (with explicit gap control to the limit).

## Caveats

Lower bound, not a matching upper-bound construction that pins a_Leb. Hierarchy exactness does not by itself produce the optimal cover shape. Numerical / certification details live in the paper body.

## Implications for Broadside

Discrete-geometry delight; pairs with covering / packing curiosities. No engine ticket, but the "monotone finite proxy → certified limit" pattern is the same shape as meshlet LOD / probe cascades that need honest error bars.

## Links

- arXiv abs: https://arxiv.org/abs/2609.01284
- PDF: https://arxiv.org/pdf/2609.01284.pdf