# Minimal Covers for Closed Curves

## Abstract

The worm problem of Leo Moser asks for the smallest plane region that can accommodate every rectifiable curve of length 1. Due to the broadness of this problem, some people choose to work on a smaller aspect of this problem. For example, instead of looking at the set of a universal curve of length 1, one can choose to work on a certain family of curves, such as a family of closed curves, a family of convex curves, etc. In this project, we study the family of closed curves. We begin by developing some theorems about 3-dimensional objects such that any closed curve of a fixed length can inscribe. Then, we use these theorems to find a smallest planar region that can accommodate any closed curve of a fixed length. Although this region is not currently the smallest cover for closed curves, it gives a great contribute to those who work on this field. We close this project by giving a status report of the Moser's worm problem regarding to the closed curves.

## Table of Contents

## Complete List of References

- [HR1] On Fenchel's theorem. American Mathematical Monthly, 78(1971) 380-381.
- [JR1] Path of minimal length within a cube. American Mathematical Monthly, 73(1966) 868-872.
- [KS1] A proof that there exists a circumscribing cube around any bounded closed covex set in R3. Annals of Mathematics, 43 (1942) 739-741.
- [NJ1] The smallest sphere containing a rectifiable curve. American Mathematical Monthly, 78(1971) 881-882.
- [SJ1] The broadest curve of length 1. University of Calgary, Department of Mathematics Research Paper No. 52.
- [WJ1] On Moser's problem of accommodating closed curves in triangles. Elem. Math., 78(1971) 35-36
- [WJ2] Triangular covers for closed curves of constant length. Elem. Math., 25 (1970) 78-82.