Lower Bounds Of Areas Of Convex Covers For Closed Unit Arcs

Moser's worm problem is the unsolved problem in geometry which asks for the minimal area of a region S on the plane which can cover all curves of unit length, assuming that curves may be rotated and translated to fit inside the region. This thesis studies a version of this problem when region S is convex and unit curves to be covered are closed. For example, region S should be able to cover a circle of length 1, a square of side length 1/4, a line interval of length 1/2, and so on. An example of such cover S is the circle of diameter 1, whose area is about 0:7854, but the problem is to find S with minimal area. Recently, Wichiramala constructed a hexagon with this property and area about 0.11023, and this is the current record. On the other hand, it is known that the area of S cannot be less than 0.096694.

In this work, we improve the lower bound for area of convex cover S for closed unit arcs from 0.096694 to 0.0975 and then to 0.1 by finding the smallest areas of convex hulls of three carefully chosen closed unit arcs. We do this by combining geometric arguments with numerical methods such as the box-search algorithm. First, we show that the minimal area of a convex hull of a circle with radius 1/2π , a rectangle with perimeter 1 and the equilateral triangle of side 1/3 is at least 0.0975. Next, we perform a systematic search for triples of closed unit arcs which leads to an even better bound. The result of our search suggests to consider circle with radius 1/2π, the rectangle with sides 0.1727 and 0.3273, and the line of length 1/2 . As the main result of this work, we prove that the minimal area of a convex hull of these three closed unit arcs is at least 0.1. This gives 0.1 as the lower bound for the area of S.