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Math-2701 · mathematics · Analysis

Uniform duality for convex-body covering entropy

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For convex bodies $K,L$ symmetric about zero, let $N(K,L)$ be the least number of translates of $L$ covering $K$, and let \[ K^\circ=\{y:\sup_{x\in K}x\cdot y\leq1\} \] be the polar of $K$, with $L^\circ$ defined similarly. Do constants $a,b\geq1$, independent of the dimension and the bodies, exist such that \[ \log N(K,L)\leq b\log N(L^\circ,a^{-1}K^\circ) \] always holds?

Short English statement

Are there constants $a,b\geq1$, independent of dimension and of the convex bodies $K,L$ symmetric about zero, such that $\log N(K,L)\leq b\log N(L^\circ,a^{-1}K^\circ)$ always holds, where $N(K,L)$ is the least number of translates of $L$ covering $K$ and $K^\circ=\{y:\sup_{x\in K}x\cdot y\leq1\}$ is the polar, defined similarly for $L$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2701-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2701

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 329; 2026-09-24).

Uniform duality for convex-body covering entropy

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 329; 2026-09-24), the asserted result is that the proposed primal--polar entropy comparison fails for any choice of universal constants, even if one covers by translates of a cube.

Mathematical statement of the claimed result

Manuscript 1.
Write $N(K,L)$ for the least number of translates of $L$ covering $K$, and $K^{\circ}$ for the Euclidean polar. The claimed counterexamples satisfy \[ \forall a,b\ge1\ \exists n\ge1\ \exists K=-K\subset\mathbb R^n:\quad \log N(K,[-1,1]^n)>b\log N(([-1,1]^n)^{\circ},a^{-1}K^{\circ}), \] where $K$ is a full-dimensional convex body. Thus no fixed pair of dilation and multiplicative constants supplies the proposed dual covering-number comparison in every dimension.

Outline of the argument

For manuscript 1 (family 329; 2026-09-24), partitions compress convex-hull columns uniformly across coordinates. Finite-field zero estimates supply many separated rows whose dual images remain compressible; choosing the matrix parameters makes the primal entropy arbitrarily larger than the proposed polar comparison.

Formal verification

For manuscript 1 (family 329; 2026-09-24), OpenAI lists Lean statements for this assertion: the proposed primal--polar entropy comparison fails for any choice of universal constants, even if one covers by translates of a cube.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are MetricEntropyDuality.lean.

Their named declarations include OAI.MetricEntropyDuality.exists_entropy_duality_counterexample_with_covers.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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