Moser's worm problem: tighten the bounds on the smallest convex cover for all unit arcs
Statement
A planar set is a cover for the family of 'worms' (rectifiable arcs of length 1) if every unit-length arc can be placed inside it by a rigid motion (translation + rotation, reflections allowed). Determine, or improve the bounds on, the minimum possible area of a convex cover for all unit arcs. Provide a smaller convex cover with an explicit covering scheme (new upper bound), or a finite family of worms whose simultaneous coverage forces a larger area (new lower bound).
Acceptance. ADVANCES (upper bound): exhibit a convex region of area strictly below the current best ($\approx 0.2709$ for convex covers) together with an explicit, checkable scheme (finite covering argument / placement map) showing every unit arc fits inside up to rigid motion. ADVANCES (lower bound): give a finite family of unit worms and a rigorous computation showing any convex region containing a congruent copy of each has area $\ge \beta$ for a new record $\beta > 0.2275$. FULLY RESOLVES: matching upper and lower bounds pinning the optimal area. Deliver the region/worm-family plus the verification program and the numeric bound achieved.
Background
Leo Moser's worm problem (c. 1966), a classic unsolved question in combinatorial/convex geometry; see Croft–Falconer–Guy, 'Unsolved Problems in Geometry', §D18, and the Moser's-worm-problem survey. The exact minimal area is unknown. For CONVEX covers the best published bounds are approximately $0.232239 \le \alpha \le 0.270911$: the upper bound $\approx 0.270911$ is Wang (2006), and the lower bound $\approx 0.232239$ is Khandhawit–Sriswasdi–Pagonakis (2013), who obtain it via a 'cage' — a finite set of specific worms (a unit segment, a V-shaped worm, a U-shaped worm) whose common cover must already have at least that area (the refined convex-cage value reported as of Oct 2024 is $\approx 0.2275$). If the cover need NOT be convex, the best known cover has area $\approx 0.260437$ (Norwood–Poole, 2003). Closely related open variants concern covers for CLOSED unit curves, where a convex cover is known to need area $\ge 0.1$ (Grechuk–Som-am and predecessors). The attacker's tool: (i) upper bounds via computational geometry — construct a candidate convex region (e.g. a trimmed rhombus/hexagon) and verify a finite covering scheme placing every extremal worm inside; (ii) lower bounds via the cage method — a finite family of worms turns 'the cover contains all of them' into a checkable area lower bound solvable by optimization. Caution: sensational claims of a 'breakthrough' on this problem circulate online; none constitutes a peer-reviewed resolution — the problem remains open.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Moser's worm problem (Wikipedia) | website |
| REF-02 | A convex cover for closed unit curves has area at least 0.1 | arxiv |
| REF-03 | A convex cover for closed unit curves has area at least 0.0975 | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.