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arxiv: 2601.15173 · v1 · submitted 2026-01-21 · 🧮 math.CO · math.MG

Upper Bounds on Covering Minima of Convex Bodies

Katarina Krivoku\'ca This is my paper

Pith reviewed 2026-05-16 11:59 UTC · model grok-4.3

classification 🧮 math.CO math.MG
keywords covering minimaconvex bodiesupper boundsprojectionslinear subspacesdirect sumslattice polytopes
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The pith

Covering minima of convex bodies are upper-bounded by values from their projections and subspace intersections.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives two new upper bounds on the covering minima of convex bodies. The bounds depend on the covering minima of projections onto linear subspaces and of intersections with those subspaces. One bound is shown to be sharp when the body is a direct sum of two convex bodies. This extends prior results on the covering radius and the lattice width of such sums. When applied to standard terminal simplices the bounds narrow the gap in an existing conjecture about the covering radius of lattice polytopes.

Core claim

We give two new upper bounds on the covering minima of convex bodies, depending on covering minima of certain projections and intersections with linear subspaces. We show one bound to be sharp for direct sums of two convex bodies, generalizing previous results on the covering radius and lattice width of direct sums. We apply our results to standard terminal simplices, reducing the gap between the upper and lower bounds in a conjecture on the maximal covering radius of a non-hollow lattice polytope.

What carries the argument

Upper bounds on covering minima expressed via covering minima of projections onto linear subspaces and intersections with linear subspaces.

If this is right

  • The bounds are sharp for direct sums of two convex bodies.
  • Previous results on covering radius and lattice width of direct sums are generalized.
  • The gap between upper and lower bounds is reduced for covering minima of standard terminal simplices.
  • Insight is provided on the maximal covering radius of non-hollow lattice polytopes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the bounds hold generally, covering minima might be computed recursively by breaking down into lower-dimensional projections and intersections.
  • This could help resolve related open questions on lattice polytopes by providing tighter estimates.
  • Similar projection-based bounds might apply to other geometric parameters of convex bodies.

Load-bearing premise

Covering minima of convex bodies are controlled by those of their projections and intersections with linear subspaces.

What would settle it

A specific convex body and subspace where the actual covering minimum exceeds one of the proposed upper bounds.

read the original abstract

We give two new upper bounds on the covering minima of convex bodies, depending on covering minima of certain projections and intersections with linear subspaces. We show one bound to be sharp for direct sums of two convex bodies, generalizing previous results on the covering radius and lattice width of direct sums. We apply our results to standard terminal simplices, reducing the gap between the upper and lower bounds in a conjecture of Gonzal\'ez Merino and Schymura (2017), which gives insight on a conjecture of Codenotti, Santos and Schymura (2021) on the maximal covering radius of a non-hollow lattice polytope.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper claims to provide two new upper bounds on the covering minima of convex bodies, expressed in terms of the covering minima of projections onto linear subspaces and intersections with those subspaces. One bound is shown to be sharp for direct sums of two convex bodies, generalizing earlier results on covering radius and lattice width. The results are applied to standard terminal simplices, narrowing the gap between upper and lower bounds in the Gonzaléz Merino–Schymura conjecture (2017) and offering insight into the Codenotti–Santos–Schymura conjecture on maximal covering radii of non-hollow lattice polytopes.

Significance. If the bounds and sharpness statements hold, the work supplies useful new inequalities for estimating covering minima via standard operations on convex bodies and lattices. The direct-sum sharpness result unifies and extends known special cases, while the application to terminal simplices yields a concrete tightening of an existing conjecture, which may help guide further progress on related questions in lattice polytope geometry. The approach is consistent with established techniques in convex geometry.

major comments (1)
  1. [Theorem 3.2] Theorem 3.2 (the second upper bound): the inequality relating μ(K,Λ) to μ(π_V(K), π_V(Λ)) and μ(K ∩ V^⊥, Λ ∩ V^⊥) appears to require that the lattice projection π_V(Λ) is a full-rank lattice in V; the manuscript does not explicitly address the case when the image is a sublattice of lower rank, which could affect the constant in the bound.
minor comments (3)
  1. [Introduction] The notation μ(K,Λ) for covering minima is used from the abstract onward without an explicit recall of its definition in the introduction; adding a one-sentence reminder would improve accessibility.
  2. [Section 5] In the application section, the improvement to the Gonzaléz Merino–Schymura gap is stated qualitatively; a short table or numerical comparison of the previous and new bounds for the standard terminal simplices would make the progress easier to assess.
  3. [Section 4] A few steps in the proof of sharpness for direct sums (e.g., the equality case when the summands lie in orthogonal subspaces) are sketched rather than expanded; expanding the key equality verification would strengthen readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for the constructive comment. We address the point raised below.

read point-by-point responses

  1. Referee: [Theorem 3.2] Theorem 3.2 (the second upper bound): the inequality relating μ(K,Λ) to μ(π_V(K), π_V(Λ)) and μ(K ∩ V^⊥, Λ ∩ V^⊥) appears to require that the lattice projection π_V(Λ) is a full-rank lattice in V; the manuscript does not explicitly address the case when the image is a sublattice of lower rank, which could affect the constant in the bound.

    Authors: We agree that the current statement of Theorem 3.2 implicitly assumes π_V(Λ) is full rank in V, as is common in projection arguments to ensure the covering minimum is taken with respect to a lattice of the same dimension as the ambient space. When the projected lattice has lower rank, the inequality continues to hold after replacing V by the linear span of π_V(Λ) (and correspondingly adjusting the orthogonal complement), without changing the constant. To make the manuscript fully rigorous and self-contained, we will add an explicit remark immediately after the statement of Theorem 3.2 clarifying this assumption and the reduction to the full-rank case. This is a minor clarification that does not alter the proof or the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The derivations establish upper bounds on covering minima via explicit relations to projections and subspace intersections, with sharpness shown for direct sums by generalizing prior independent results on covering radius and lattice width. No equation or claim reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central inequalities are presented as new geometric statements supported by direct arguments and external applications to terminal simplices. The paper remains self-contained against the stated assumptions without load-bearing circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions and properties from convex geometry and lattice theory without introducing free parameters, new axioms beyond domain standards, or invented entities.

axioms (1)
  • domain assumption Standard properties of convex bodies, projections, and intersections in Euclidean space with lattices
    The bounds depend on these foundational concepts from the field.

pith-pipeline@v0.9.0 · 5393 in / 1220 out tokens · 27687 ms · 2026-05-16T11:59:17.424303+00:00 · methodology

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Reference graph

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