Hadwiger's conjecture for cap bodies

Hadwiger’s covering conjecture is that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic copies, with $2^n$ copies required only for affine images of $n$-cube. Convex hull of a ball and an external point is called a spike. The union of finitely many spikes of a ball is a cap body if it is a convex set. In this note, we confirm the Hadwiger’s conjecture for the class of cap bodies in all dimensions, bridging recently established cases of $n=3$ and large $n$. The proof uses probabilistic techniques, and additionally, for moderate dimensions $4\le n \le 15$, integer linear programming performed with computer assistance.

💡 Research Summary

This paper presents a complete proof of Hadwiger’s Covering Conjecture for the class of “cap bodies” in all dimensions. Hadwiger’s conjecture, a longstanding open problem in discrete geometry, states that every n-dimensional convex body can be covered by at most 2^n smaller positive homothetic copies of itself, with equality only for affine images of the n-cube. The conjecture is equivalent to an “illumination” problem: finding the smallest number of external light sources (directions) that can illuminate every point on the boundary of a convex body.

The authors focus on cap bodies, a specific family of convex sets defined as the convex hull of a finite number of “spikes” attached to a unit ball. A spike is the convex hull of the unit ball and a single point outside it. Each spike corresponds to a spherical cap on the ball’s surface. The class of n-dimensional cap bodies is denoted by K_nc.

The main results are divided into three parts, addressing different dimensional ranges:

1. For dimensions 4 ≤ n ≤ 15, the authors prove Theorem 1 using a combination of probabilistic methods and computer-assisted integer linear programming (ILP). The illumination directions are chosen as the vertices of several randomly rotated regular simplices and/or randomly rotated cross-polytopes. The key idea is to estimate the expected number of spherical caps (from the cap body) that remain unilluminated by these random sets. This expectation is bounded by solving an ILP problem, where the variables represent the number of caps of certain sizes, subject to constraints derived from the non-overlapping property of the caps and a fundamental spherical packing bound (Lemma 8). The solution to this ILP provides an upper bound on the number of additional directions needed to fully illuminate any cap body. Specific upper bounds strictly smaller than 2^n are listed in Table 1 (e.g., at most 11 directions for n=4, 69 for n=8).

2. For large dimensions (n ≥ 9), the authors prove Theorem 2 using a more analytical, probabilistic approach without computer assistance. Here, only random rotations of the cross-polytope are used as direction sets. By classifying caps based on their size and using explicit inequalities for the area of spherical caps (Equation (3)), the authors derive an upper bound for the expected number of unilluminated caps as a function of the number of random rotations. Minimizing this function analytically yields an explicit upper bound for the illumination number (Equation (1)). This bound guarantees that I(K_nc) < 2^n for all n ≥ 13.

3. Combining these results with the previously established case for n=3 (where I(K_3c)=6), the authors conclude in Corollary 3 that Hadwiger’s conjecture holds true for the entire class of cap bodies in all dimensions n ≥ 3. A crucial observation is that for n ≥ 3, the class of cap bodies does not contain any affine image of an n-cube, which is the only case requiring the full 2^n illuminations according to the conjecture.

The paper’s methodology is noteworthy for its interdisciplinary blend of spherical geometry, probabilistic combinatorics, and computational optimization. The ILP-based proof for moderate dimensions provides concrete, verifiable bounds, while the analytical proof for high dimensions offers asymptotic insight. The appendix includes a SageMath script to verify the ILP computations, ensuring reproducibility. This work resolves a significant special case of Hadwiger’s conjecture, bridging the gap between recently solved low-dimensional and asymptotic results.