Lebesgue Minimal Problem

Find the plane Lamina of least Area $A$ which is capable of covering any plane figure of unit General Diameter. A Unit Circle is too small, but a Hexagon circumscribed on the Unit Circle is too large. More specifically, the Area is bounded by

$$0.8257\ldots = {\textstyle{1\over 8}}\pi+{\textstyle{1\over 4}}\sqrt{3}<A<{\textstyle{2\over 3}}(3-\sqrt{3}\,)=0.8452\ldots$$

(Pal 1920).

See also Area, Borsuk's Conjecture, Diameter (General), Kakeya Needle Problem

References

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

Coxeter, H. S. M. ``Lebesgue's Minimal Problem.'' Eureka 21, 13, 1958.

Grünbaum, B. ``Borsuk's Problem and Related Questions.'' Proc. Sympos. Pure Math, Vol. 7. Providence, RI: Amer. Math. Soc., pp. 271-284, 1963.

Kakeya, S. ``Some Problems on Maxima and Minima Regarding Ovals.'' Sci. Reports Tôhoku Imperial Univ., Ser. 1 (Math., Phys., Chem.) 6, 71-88, 1917.

Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 142-144, 1990.

Pál, J. ``Ueber ein elementares Variationsproblem.'' Det Kgl. Danske videnkabernes selskab, Math.-fys. meddelelser 3, Nr. 2, 1-35, 1920.

Yaglom, I. M. and Boltyanskii, V. G. Convex Figures. New York: Holt, Rinehart, & Winston, pp. 18 and 100, 1961.