A claimed failure of universal entropy duality

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Family 329 · Functional analysis
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

How many small shapes are needed to cover a larger shape? This elementary-sounding question is a way to measure complexity: each translate in a cover represents a collection of points that need not be distinguished at the chosen resolution.

For symmetric convex bodies, there is a natural duality operation called polarity. It exchanges a body with the set of linear measurements bounded by one on that body. Pietsch’s entropy-duality conjecture asks whether covering complexity is preserved, up to universal constants, when one reverses the roles of two bodies and takes their polars.

Family 329 claims a negative answer to the unrestricted, dimension-free conjecture. For every proposed pair of universal constants, it constructs bodies violating the required comparison. The covering body can already be an ordinary cube.

The result does not say that polarity destroys every useful comparison. Strong positive theorems remain available for ellipsoids, geometrically controlled spaces, and comparisons with dimension-dependent losses. The issue is whether one fixed pair of constants works for all symmetric convex bodies in every dimension.

2. Background and essential definitions

A convex body in $\mathbb R^n$ is a compact convex set with nonempty interior. It is origin-symmetric when $x$ belongs exactly when $-x$ does. For bounded sets $A,B$, with $B$ having interior, the covering number $N(A,B)$ is the smallest number of translates of $B$ whose union contains $A$. The centers of the translates can lie anywhere in the ambient space.

The quantity $\log N(A,B)$, using natural logarithms, is called covering entropy. Taking the logarithm turns multiplicative increases in the number of approximations into additive increases in complexity.

For an origin-symmetric convex body $K$, its polar is

$$K^\circ=\{y:\ |\langle x,y\rangle|\leq1 \text{ for every }x\in K\}.$$

The absolute value can be omitted for symmetric $K$. A larger body has a smaller polar. The polar of the cube $[-1,1]^n$ is the unit ball $\{y:\sum_i|y_i|\leq1\}$.

The conjectured comparison asks for absolute constants $a,b\geq1$ satisfying

$$b^{-1}\log N(L^\circ,aK^\circ) \leq\log N(K,L) \leq b\log N(L^\circ,a^{-1}K^\circ)$$

for every dimension and every pair $K,L$. The constant $a$ permits a fixed change of resolution; $b$ permits a fixed multiplicative change in entropy. Both allowances are important: the question is much weaker than equality of the two covering numbers.

3. Development history and previous partial results

The problem originated in Pietsch’s 1972 work on entropy numbers of linear operators. These numbers measure how efficiently the image of a unit ball can be approximated by finitely many balls. Passing from an operator to its adjoint leads to the geometric polar-body formulation.

Early results already established substantial dual control. König and Milman compared the appropriate roots of covering numbers, allowing an additive error proportional to dimension on the logarithmic scale. Bourgain, Pajor, Szarek, and Tomczak-Jaegermann obtained estimates under geometric or sequence-regularity assumptions, and estimates involving rank-dependent losses.

Artstein, Milman, and Szarek proved the dimension-free conjecture when one body is a Euclidean ball, hence also when one body is an ellipsoid. This settled a central Hilbert-space case. It did not cover arbitrary unit balls of arbitrary norms.

Artstein, Milman, Szarek, and Tomczak-Jaegermann also introduced convexified packing and proved a universal duality theorem for that quantity. In an ordinary packing, points are separated pairwise. In convexified packing, each new translated neighborhood must avoid the convex hull of its predecessors. This ordered requirement is stronger and changes the quantity being counted.

The same work gave ordinary entropy duality under a bound on a K-convexity constant, which measures a space’s random-sign geometry. Emanuel Milman later obtained comparisons for general symmetric bodies with logarithmic losses in scale and entropy. These results identify important positive regimes but leave room for failure of one universal pair of constants across all dimensions.

4. What the new manuscript claims

M1 states that for every $a,b\geq1$, there are an integer $n\geq1$ and an origin-symmetric convex body $K\subset\mathbb R^n$ such that, with $L=[-1,1]^n$,

$$\log N(K,L) >b\log N(L^\circ,a^{-1}K^\circ).$$

Thus the upper inequality of the conjecture fails for every proposed universal pair. That is sufficient to disprove the two-sided formulation. The construction gives bodies with nonempty interior in their full ambient dimension; it is not exploiting a lower-dimensional set whose polar behaves degenerately.

The dimension and body depend on the proposed constants. There is no claim of a fixed dimension in which all possible constants fail. The quantifier order is the point: choose the constants first, and the construction supplies a sufficiently demanding example afterward.

M1 also derives a separation between ordinary covering entropy and convexified-packing entropy, even after a fixed change of scale. This explains why the earlier universal convexified-packing theorem need not extend to ordinary covering numbers.

The manuscript formulates its main result for arbitrary translation centers, the usual definition of covering number. Its compression construction therefore has to account carefully for whether approximation centers belong to the set being approximated; the geometric conversion includes that bookkeeping.

5. Why the advance matters

Importance: 5/5. The family claims a counterexample to a central universal duality conjecture in the local theory of Banach spaces and convex geometry. A negative resolution would substantially change the boundary between useful dual comparisons and comparisons that demand additional hypotheses.

The cube makes the statement particularly striking. One need not choose two unfamiliar bodies: even covering by a standard coordinate box can display the alleged failure. The complexity is carried by the other body and the dimension-dependent construction.

The result also separates notions that can initially appear interchangeable. Pairwise distinguishability, ordered convexified packing, and approximation by a small list all express complexity, but they need not obey the same duality principles. Understanding these distinctions matters when transferring geometric results into approximation or learning problems.

At the same time, the proposed counterexample leaves the ellipsoid theorem and bounded-geometric-constant results intact. Family 326’s conditional entropy consequence has precisely such additional geometric control, so its positive statement and this negative one concern different ranges of hypotheses.

6. The main idea

The construction begins with a finite matrix whose entries lie between zero and one. Think of rows as objects to distinguish and columns as measurements on those objects. M1 seeks two properties that pull in opposite directions: many rows are well separated in the maximum-coordinate norm, while all signed convex combinations of columns admit a short uniform approximation list.

A support-function calculation translates this discrepancy into convex bodies. The support function of a body assigns to each linear measurement its largest value on that body. Taking a symmetric hull of the rows creates large primal covering complexity, while the compressed column combinations control the polar covering complexity. A small cube is added to ensure nonempty interior.

The column compression uses a finite collection of partitions of a set. Selected partitions define a graph: two points are linked when they share a class in one of those partitions. Columns are profiles that decrease logarithmically with graph distance from a chosen center and become zero beyond a cutoff.

For a convex combination of columns, averaging identifies a partition used by most of the coefficient mass. At each distance scale, a greedy choice of a few pivots captures nearly all relevant mass. A pivot can then be replaced by a fixed representative of its class in that common partition.

This replacement is the compression step. One stores a class label instead of an arbitrary original point. Although replacement enlarges graph-distance radii, the logarithmic profile makes the resulting error uniformly small. Positive and negative coefficient parts are treated separately to cover the absolutely convex hull, meaning the convex hull of the columns and their negatives.

The separated rows come from symmetric multilinear forms over a finite field. A symmetric $h$-linear form takes $h$ vectors as input, is linear in each, and is unchanged when inputs are permuted. Carefully chosen directions ensure a nonvanishing condition after deleting a small fraction of them. A short graph path would contradict this condition, forcing different rows to be separated.

The numerical advantage comes from the different sizes of the forms and their contracted labels. With $r$ field coordinates, write

$$D_j=\binom{r+j-1}{j}.$$

There are $p^{D_h}$ possible rows but at most $p^{D_{h-1}}$ labels per partition, where $p$ is a sufficiently large prime. Their logarithmic size ratio satisfies

$$\frac{D_{h-1}}{D_h}=\frac{h}{r+h-1}.$$

For fixed $h$, this tends to zero as $r$ increases. The remaining encoding costs are fixed before $p$ is chosen, so a large prime makes those costs negligible. This ordered choice of parameters turns the row-versus-column discrepancy into a violation of any prescribed universal constants.

7. Scope, limitations, and open questions

The claim concerns finite-dimensional real symmetric convex bodies, but dimensions are allowed to grow. It does not contradict the existence of constants depending on dimension, nor comparisons with logarithmic or geometric losses.

The counterexample is an existence construction. The main statement does not advertise a small, practical-dimensional pair of bodies or an efficient numerical procedure for finding the first failure for given constants. Its purpose is to defeat uniformity, not to optimize the size of a counterexample.

The result also does not identify the best possible replacement for the conjecture. Determining sharp dimension-dependent losses, finding broader positive geometric classes, and clarifying exactly when ordinary and convexified entropy remain comparable are separate questions.

The manuscript’s definitions must be kept fixed when comparing these claims: covering uses arbitrary translates, the bodies have full interior, and the polar pair reverses their order. The construction and its consequences are reported as stated, without a proof audit.

8. Sources and further reading

• Artstein, Milman, and Szarek, Duality of metric entropy, for the Euclidean-ball and ellipsoid case.
• Artstein, Milman, Szarek, and Tomczak-Jaegermann, On convexified packing and entropy duality.
• E. Milman, A remark on two duality relations, for comparisons with losses.
• Optional background: polar sets.

Source manuscripts

• M1. OpenAI, Counterexamples to the duality conjecture for metric entropy (September 24, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.