Convex Bodies of Constant Width

Geometric Inequalities for Plane Hedgehogs

“…For a study of projective hedgehogs in E" +1 , see [3]. A survey of convex bodies of constant width is given by Chakerian and Groemer [1], As a corollary, we have the following result.…”

“…We may consider C as the envelope of the family of support lines given by (1) x cos 0 y sin 0 = p(0), where the support function p(0) = h(cos6, sin0) is defined as the signed distance of the support line to C with exterior normal vector u(6) = (cos 0, sin 9) from the origin. Given any h € C 2 (S 1 ;R), we may always consider the envelope Hh of the family of lines given by (1).…”

“…We may consider C as the envelope of the family of support lines given by (1) x cos 0 y sin 0 = p(0), where the support function p(0) = h(cos6, sin0) is defined as the signed distance of the support line to C with exterior normal vector u(6) = (cos 0, sin 9) from the origin. Given any h € C 2 (S 1 ;R), we may always consider the envelope Hh of the family of lines given by (1). Partial differentiation of (1) yields (2) -xsin0 + ycos0 = p'(0), and from (1) and (2) We say that Hh is the hedgehog defined by the support function h. In general, Hh is not a convex curve of class C 1 .…”

Two dimensions are easier

“…When Ω is a cube, no explicit analytical description of its Cheeger set has been given, other than that it is convex and that those parts of its boundary that do not touch the cube have constant mean curvature |ΩC | |∂ΩC | . A numerical approximation and visualization can be found in [11] Open Problem 2: Convex sets of constant width [2,3,14] have been studied for more than a century. A nice exposition can be found in the book "Geometry and the Imagination" by Hilbert and Cohn-Vossen [10].…”

Adjoint transform, overconvexity and sets of constant width

“…Before doing so, we shall recall that a convex set K is said to be of constant width d if its support function satisfies everywhere h(<f>) + h(n + <p) = d. This amounts (in the euclidean case) to say that K is complete, that is such that the addition of any point to K strictly increases its diameter. (See Chakerian and Groemer [9] for a recent and fairly exhaustive survey of properties of sets of constant width.) Finally, a set Kc is said to be a completion of K if K ç Kc, d(K) = d(Kc) and Kc is of constant width.…”