Kakeya Needle Problem

What is the plane figure of least Area in which a line segment of width 1 can be freely rotated (where translation ofthe segment is also allowed)? Besicovitch (1928) proved that there is no Minimum Area. This can be seenby rotating a line segment inside a Deltoid, star-shaped 5-oid, star-shaped 7-oid, etc. When the figure is restrictedto 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

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.

• Dirichlet Series
• Dirichlet's Formula
• Dirichlet's Lemma
• Dirichlet's Principle
• Dirichlet's Test
• Dirichlet's Theorem
• Dirichlet Structure Constant
• Dirichlet Tessellation
• Dirty Beam
• Dirty Map
• Disc
• Disconnected Form
• Disconnectivity
• Discontinuity
• Discontinuous
• Discordant Permutation
• Discrepancy Theorem
• Discrete Distribution
• Discrete Fourier Transform
• Discrete Mathematics
• Discrete Set
• Discrete Uniform Distribution
• Discriminant
• Discriminant (Binary Quadratic Form)
• Discriminant (Elliptic Curve)