The optimal order of convex-body covering density

1 · The essentials

How much overlap is unavoidable when copies of one shape must cover an entire space without gaps? The shapes considered here have no inward dents, and their copies may slide but cannot turn or change size. The main manuscript says that, in every dimension $n \geq 2$ , copies arranged in one regular repeating pattern can cover space with density at most $C n lo g n$ , where $C$ is a fixed constant independent of the shape and dimension. Density measures how much material the covering uses per unit of space, counting overlaps. A companion manuscript shows that, in every sufficiently large dimension, some shapes require density greater than $c n lo g n$ even when the copies can be placed without a repeating pattern. Together, these bounds identify the worst-case growth rate, though not an exact value. The main covering bound has a Lean formalization; the companion lower bound is not identified as formalized in the supplied source.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine covering a tabletop with identical paper shapes that you may slide but not rotate, leaving no gaps. The question is how much extra paper must pile up in overlaps, and whether placing the shapes in a regular repeating pattern makes the task harder. The actual results concern covering an entire space in arbitrary dimensions, not a finite tabletop.

Why it matters

The two manuscripts give matching worst-case growth rates for coverings with and without a lattice restriction. The companion examples rule out a bound proportional only to dimension that would work for every convex body. The main result achieves the matching upper rate using just one full-rank lattice, without assuming symmetry or boundary regularity.

2 · Going further

The problem is to control the density of coverings of $R^{n}$ by translates of a convex body. In a lattice covering, the translation vectors must belong to one full-rank lattice; an unrestricted translative covering has no such requirement. The main manuscript gives an absolute constant $C > 0$ such that every convex body in every dimension $n \geq 2$ admits a lattice covering of density at most $C n lo g n$ . No symmetry or boundary regularity is assumed.

The companion manuscript shows that, for every sufficiently large $n$ , there is a centrally symmetric convex body whose translative covering density exceeds $c n lo g n$ , for an absolute constant $c > 0$ . Because this lower bound applies even without a lattice restriction, it also limits what lattice coverings can achieve for those bodies. Combined with the main upper bound, it yields worst-case order $Θ (n lo g n)$ in both settings.

The supplied abstract describes the companion's approach as constructing centrally symmetric examples, but gives no construction details. The main abstract states the single-lattice bound without describing its argument, so no further proof mechanism can be extracted from the source. For context, the companion says its lower bound matches the order of Rogers' upper bound and disproves a universal linear upper bound. The main manuscript is marked Lean-formalized; that verification status is not stated for the companion.

Useful notions

Convex bodies and their translates
Coverings of Euclidean space
Covering density
Full-rank lattices
Asymptotic growth up to constant factors

What was known before / what is new

Before

The companion abstract identifies Rogers' upper bound as having the order matched by its lower bound. The supplied source gives no further historical account of earlier lattice bounds or of the status of a universal linear upper bound before these manuscripts.

New

The main manuscript supplies a single-lattice upper bound of $C n lo g n$ for every convex body in dimension $n \geq 2$ , with no symmetry or boundary-regularity assumption. The companion manuscript shows that centrally symmetric examples in every sufficiently large dimension require more than $c n lo g n$ even for unrestricted translative coverings. Together, these statements determine the optimal worst-case order for both types of covering.

Limits and assumptions

The upper bound concerns convex bodies in $R^{n}$ with $n \geq 2$ ; the lower-bound examples are asserted only for sufficiently large dimensions. The lower bound is a worst-case existence statement, not a requirement for every convex body. The constants are absolute, but their values and an exact worst-case density are not supplied. The main single-lattice upper bound has a Lean formalization; the source does not identify the companion lower bound as formalized or give further details about formalization scope. The supplied text lists no remaining open problems.

Mini glossary

Convex body: A compact shape with nonempty interior that contains the line segment between any two of its points.
Translative covering: A covering of space by copies of a body moved only by translation, without rotation or resizing.
Covering density: The average amount of volume contributed by covering copies per unit volume of space, counting overlaps.
Full-rank lattice: A repeating set of points generated by integer combinations of a basis of the ambient space.
Centrally symmetric: A body is centrally symmetric if reflection through a central point leaves it unchanged.
$Θ (n lo g n)$: Growth bounded above and below by positive constant multiples of $n lo g n$ for sufficiently large $n$ .

Keywords : Convex bodies · Covering density · Lattice coverings

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)

Determines the optimal worst-case covering density as $Θ (n lo g n)$ , for both lattice and unrestricted translative coverings. Every convex body in ℝⁿ, n ≥ 2, admits a lattice covering of density at most $C n lo g n$ ; centrally symmetric examples in every sufficiently large dimension require at least $c n lo g n$ even without the lattice restriction, for absolute $c, C > 0$ .

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.

Scope of the Lean formalization (source text)

…

lean/docs/092.md

Manuscripts in this family

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A single-lattice covering bound of order n log n

23/09/2026

Verified by Lean

Every convex body in ℝⁿ, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most $C n lo g n$ , for an absolute constant C. No symmetry or boundary regularity is assumed.

PDF GitHub folder Lean formalization

Translative covering densities of order n log n

23/09/2026

To be verified

For every sufficiently large dimension n, we construct a centrally symmetric convex body whose translative covering density exceeds $c n lo g n$ , where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.

PDF GitHub folder