Hadwiger's illumination / covering problem in R^3: beat the bound of 14
Statement
The illumination number $I(K)$ of a convex body $K\subset\mathbb{R}^3$ is the least number of external light sources (equivalently, directions) needed so that every boundary point of $K$ is illuminated; equivalently, the least number of smaller positive homothets of $K$ whose union covers $K$. Hadwiger's conjecture asserts $I(K)\le 2^3 = 8$ for every convex body in $\mathbb{R}^3$, with equality only for parallelepipeds. Improve the best known general upper bound (currently $14$) toward $8$, or settle the conjecture in $\mathbb{R}^3$.
Acceptance. ADVANCES: prove $I(K)\le m$ for all convex bodies $K\subset\mathbb{R}^3$ with some $m\le 13$ (strictly below the current record 14), via an argument whose combinatorial/covering core is machine-checkable — e.g. an explicit finite direction set per body class plus a verified boundary-illumination certificate. FULLY RESOLVES: prove $I(K)\le 8$ for all convex bodies in $\mathbb{R}^3$ (Hadwiger in dimension 3), or exhibit a convex body with $I(K) > 8$ together with a rigorous lower-bound certificate (a witness set of boundary points no admissible small-homothet/direction system can jointly cover). Deliver the configuration + verifier and the bound obtained.
Background
The Levi–Hadwiger–Boltyanski illumination conjecture: every convex body $K\subset\mathbb{R}^n$ satisfies $I(K)\le 2^n$, with equality iff $K$ is an affine cube. It is a theorem for $n=1,2$ ($I\le 4$ in the plane, Levi 1955), but OPEN for every $n\ge 3$. In $\mathbb{R}^3$ the conjectured value is $8$; the best known general upper bound is $14$ (Prymak, improving the long-standing $16$ of Papadoperakis and earlier Rogers-type bounds). Progress exists for special classes: centrally symmetric bodies, bodies of constant width (conjectured $4$, known $\le 6$), zonotopes, and cap bodies (recently shown to need only $6$) — but the general body in $\mathbb{R}^3$ is unresolved. Surveys: Bezdek–Khan, 'The geometry of homothetic covering and illumination' (arXiv:1602.06040); Brass–Moser–Pach, 'Research Problems in Discrete Geometry'. The attacker's tool: illumination/covering is a configuration problem — a proposed bound is certified by exhibiting, for a worst-case family of bodies, an explicit finite set of directions (or homothety centers + ratios $<1$) and verifying the union covers the boundary. Computational attack routes: reduce to covering the unit sphere of normal directions by 'illuminated caps', use SDP/LP relaxations, or search over candidate direction sets with a rigorous boundary-coverage verifier.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Hadwiger conjecture (combinatorial geometry) — Wikipedia | website |
| REF-02 | Bezdek & Khan — The geometry of homothetic covering and illumination | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.