∑math20

Functional analysis · Result 329

A counterexample to metric-entropy duality

1 paperLean

Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 1

First page of Counterexamples to the duality conjecture for metric entropy

Counterexamples to the duality conjecture for metric entropy

OpenAI · September 24, 2026

We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube.

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@misc{OAI:Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026,
  author = {{OpenAI}},
  title = {{Counterexamples to the duality conjecture for metric entropy}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf}{OAI:Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Metric-entropy duality predicts a universal comparison between covering numbers of convex bodies and their polars. The formalized result disproves such a comparison: for every \(a,b\ge1\), there is an origin-symmetric convex body \(K\) in a positive finite dimension with \(\log N(K,B_\infty)>b\log N(B_\infty^\circ,a^{-1}K^\circ)\), where \(N\) is the least number of translates in a finite cover. A further construction makes the dual-to-primal logarithmic entropy ratio tend to zero as the dimension grows.

Comparator links

ResultComparator statement
Counterexample to metric-entropy dualityMetricEntropyDuality.lean

Scope note on GitHub →