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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
See also Curve of Constant Width, Lebesgue Minimal Problem, Reuleaux Polygon, Reuleaux Triangle
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.