What is the plane figure of least Area in which a line segment of width 1 can be freely rotated (where translation of
the segment is also allowed)? Besicovitch (1928) proved that there is *no* Minimum Area. This can be seen
by rotating a line segment inside a Deltoid, star-shaped 5-oid, star-shaped 7-oid, etc. When the figure is restricted
to be convex, Cunningham and Schoenberg (1965) found there is still *no* minimum Area. However, the smallest *simple convex* domain in which one can put a segment of length 1 which will coincide with itself when rotated by 180°
is

(Le Lionnais 1983).

**References**

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.* New York: Dover, pp. 99-101, 1987.

Besicovitch, A. S. ``On Kakeya's Problem and a Similar One.'' *Math. Z.* **27**, 312-320, 1928.

Besicovitch, A. S. ``The Kakeya Problem.'' *Amer. Math. Monthly* **70**, 697-706, 1963.

Cunningham, F. Jr. and Schoenberg, I. J. ``On the Kakeya Constant.'' *Canad. J. Math.* **17**, 946-956, 1965.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 24, 1983.

Ogilvy, C. S. *A Calculus Notebook.* Boston: Prindle, Weber, & Schmidt, 1968.

Ogilvy, C. S. *Excursions in Geometry.* New York: Dover, pp. 147-153, 1990.

Pál, J. ``Ein Minimumproblem für Ovale.'' *Math. Ann.* **88**, 311-319, 1921.

Plouffe, S. ``Kakeya Constant.'' http://www.lacim.uqam.ca/piDATA/kakeya.txt.

Wagon, S. *Mathematica in Action.* New York: W. H. Freeman, pp. 50-52, 1991.

© 1996-9

1999-05-26