Constant width, body of

A convex body for which the distance between any pairs of parallel supporting planes is the same. This distance is called the width of the body. Besides closed balls there exist infinitely many other, generally speaking non-smooth, bodies of constant width. The simplest of them is the body bounded by the surface obtained by rotating a Reuleaux triangle around one of its axes of symmetry. The class of bodies of constant width coincides with the class of convex bodies of constant perimeter, for which the boundaries of the orthogonal projections on all possible planes have the same lengths.

Apart from bodies of constant width one sometimes considers bodies of constant brightness, which are characterized by the fact that the area of their orthogonal projections on planes is constant.

References

[1]	W. Blaschke, "Kreis und Kugel" , Chelsea, reprint (1949)

Comments

Two important characterizations of convex bodies of constant width are: 1) a convex body is of constant width if and only if each normal to its boundary is a double normal; and 2) a set in Euclidean space is complete if and only if it is a convex body of constant width. Here, a set $S$ in Euclidean space is called complete if its diameter must increase when a single point is added to $S$.

For a generalization of constant width due to S.A. Robertson see Constant width, curve of (especially references [2] and [a2]); that article also contains a definition of Reuleaux triangle.

References

[a1]	T. Bonnesen, W. Fenchel, "Theorie der konvexen Körper" , Springer (1934)