Moser’s convex worm problem

A straight curve of length one and a right-angle bend with two half-length arms can fit in the same unit square, each placed separately. How small an area can one convex container have while fitting every length-one planar curve, however it bends?[1] [2]

How little planar area can a convex container use if it must fit every curve of length one? A unit segment and the path $(0,0)\to(1/2,0)\to(1/2,1/2)$ both have length one and fit in a unit square, each under its own placement. These two examples illustrate the rule, not a proposed optimal cover.

A rectifiable curve is continuous and has finite travel length; self-intersections and retracing are allowed, with retraced travel contributing to length. The container holds the curve’s trace after a translation, rotation or reflection. Convexity requires it also to contain segments between any two of its points.[1] [2]

The best possible region and its exact area remain unknown.[2]

The target infimum is the greatest lower bound on areas of all convex universal covers. Existence of a minimizing convex cover is known in the cited treatments; its geometry and value are not. The September preprint reports computer-assisted lower and upper bounds with a remaining gap.[1] [2]

Length measures how far a curve travels; area measures how much room surrounds it. This problem compares those two kinds of size across all shapes with the same travel length.[1]

Convexity turns the directions visited by a one-dimensional path into a two-dimensional footprint. Equal travel lengths can produce quite different convex hulls. A universal container compares those geometric demands simultaneously, giving a concrete link between length, bending and area rather than optimizing for one chosen curve.[1] [2]