On the girth of convex bodies

Abstract

For a three-dimensional convex body and a given direction the corresponding girth is defined as the perimeter of the orthogonal projection of the body onto a plane orthogonal to the assigned direction. Obvious examples show that there are pairs of convex bodies that have in every direction equal girths but are not translates of each other. Here a modification of the concept of the girth, called semigirth, is introduced, and it is shown that convex bodies are uniquely determined (up to translations) by their semi-girths. In addition, corresponding stability results are proved and an inequality concerning the central symmetry of convex domains is established.Let K be a convex body. In the present context that means a nonempty compact convex subset of euclidean three-dimensional space 3 . Let e denote the class of all planes in 3 that contain the origin o of 3 . For any E 2 e we write K E for the orthogonal projection of K onto E. The compact convex subsets of E will be referred to as convex domains, and the perimeter of a convex domain D will be denoted by pD. Then pK E is called the girth of K with respect to E. It is well-known that the girth of a convex body, considered as a function of E, does not uniquely determine the body. (Here and in the sequel, "uniquely" is to be understood as "uniquely up to translations".) A simple example that shows this lack of uniqueness is provided by comparing a ball of diameter d with a non-spherical convex body K of constant width d. On the other hand, the knowledge of the girth of a convex body does provide some information on the body itself. A striking result in this direction is the following theorem of Minkowski [12]: If the girth of a convex body K (viewed as a function on e ) is constant, then K has constant width. For further results and references concerning this and related results on convex bodies of constant width see the survey article [3]. A more general result than Minkowski's was proved by Nakajima [13]. To formulate it recall that the width of K in the direction u, denoted by w K u, is defined as the distance between the two parallel support planes of K that are orthogonal to u. Two convex bodies K and L are said to be equiwide if for every direction w K u w L u. Nakajima's result can be formulated as follows: If K and L are two convex bodies such that pK E pL E (for all E 2 e ), then K and L are equiwide. Actually the proof of this theorem requires only slight modifications of the original proof of Minkowski which depends on the expansion of the support function of a convex body as a series of spherical harmonics. Details and substantial generalizations of this

Fourier transforms and the Funk-Hecke theorem in convex geometry

“…In recent times there has been a great deal of interest in obtaining similar results in the absence of the symmetry condition. We mention, for example, the work of Groemer [12,13], Goodey and Weil [8,9], Schneider [28] and Böröczky and Schneider [1]. Here, we use Fourier transforms and the Funk-Hecke theorem to obtain further results, similar in nature to those of Schneider and of Böröczky and Schneider.…”

“…The real part off −1 was established in [18] and used to provide a Fourier transform solution to the Shephard problem [29]. The most significant feature of the imaginary part off 0 is the hemispherical transform, which plays an important role in some of Schneider's geometrical applications [26] and in Groemer's work [12,13]. Its real part was calculated by Yaskin and Yaskina in their work on centroid bodies [31].…”

Spherical projections and liftings in geometric tomography

“…The Blaschke section bodies of the second kind (Example 6) and the k-th support bodies with weight 1 − k (Example 7) both correspond to the operator π . Building on the ideas in [17] and [18], these were studied in [13] and [14] where equation 8.6 was used to show that π…”

The determination of convex bodies from the size and shape of their projections and sections

“…We will also find some results which fail in certain exceptional dimensions. In §6, we will see how the work of Groemer [24] and [25] can be put into the general setting of directed data. Finally, §7 will address some of the possible pairings of data regarding sections and projections that can be used to determine a body uniquely.…”