1 · The essentials
How many copies of one shape are needed to cover another? The conjecture discussed here proposed that this question could be compared with a corresponding question about associated “dual” shapes, using fixed numerical allowances that work in every dimension. The manuscript gives counterexamples to that proposal. Whatever pair of universal constants is proposed, suitable shapes violate the required comparison. These counterexamples use convex shapes symmetric about the origin, and the shape used for covering can even be a cube. The main result has a Lean formalization. The conclusion concerns this particular universal comparison, not every possible relationship between covering questions and dual shapes.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine covering a tabletop with identical tiles and hoping that a fixed rule always compares the number needed with the number needed for a related tabletop-and-tile pair. The counterexample says that any proposed pair of fixed allowances can fail; the image does not capture how the mathematical dual shapes are formed or how dimension enters.
Why it matters
The result rules out Pietsch’s proposed dimension-free duality comparison for metric entropy. Failure already occurs when the covering body is a cube, so this restriction does not rescue the conjecture. It marks a limit on what a universal comparison of these covering quantities can assert.
2 · Going further
The problem concerns covering numbers of origin-symmetric convex bodies and a conjectured comparison with covering numbers involving their polar bodies. Here, origin symmetry means that a body contains whenever it contains . Pietsch’s dimension-free conjecture asks for a pair of universal constants in the covering-entropy inequality: the same constants must work independently of dimension. The supplied abstract does not state the exact inequality.
The main result reverses those universal requirements: for every proposed pair of constants, there are origin-symmetric convex bodies violating the corresponding inequality. The covering body can already be a cube. This is the counterexample strategy described in the abstract; no details of the construction or estimates are supplied, so a more specific account of the argument is not available from this source.
The supplied family contains one manuscript, “Counterexamples to the duality conjecture for metric entropy,” whose main result has a Lean formalization. No companion manuscripts are supplied, so no relationship to additional results can be described. The source also does not give a more detailed account of the formalization’s scope.
Useful notions
- Convex bodies and origin symmetry
- Covering numbers
- Polar bodies
- Universal constants and dimension dependence
- Counterexamples to universal statements
What was known before / what is new
Before
The source identifies Pietsch’s dimension-free duality conjecture for metric entropy as the proposal being addressed. It sought a universal comparison between covering numbers and the corresponding quantities for polar bodies. The source supplies no further history or earlier partial results.
New
The manuscript supplies counterexamples for every proposed pair of universal constants in the conjectured inequality. It does so even with a cube as the covering body. The main counterexample result has a Lean formalization.
Limits and assumptions
The stated result concerns Pietsch’s dimension-free covering-entropy comparison for origin-symmetric convex bodies. It defeats every proposed pair of universal constants, including under the restriction that the covering body is a cube; it does not assert that every other form of duality comparison fails. The main result has a Lean formalization, but the supplied source gives neither the exact formal statement nor details of its coverage beyond that designation. No explicit construction, exact inequality, companion results, or remaining open questions are provided.
Mini glossary
- Covering number —
- A quantity recording how many copies of a covering body are needed to cover another body.
- Origin symmetry —
- The property that a body contains the point whenever it contains .
- Polar body —
- The associated dual body used in the conjectured comparison of covering quantities.
- Dimension-free —
- Describes a comparison whose constants do not depend on the dimension.
- Universal constants —
- Fixed numerical values required to work for every body and dimension covered by the conjecture.
Keywords : Metric entropy · Polar bodies · Dimension-free duality
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Counterexamples to the duality conjecture for metric entropy
24/09/2026We 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.