Shaking Compact Sets

If C is a compact subset of R and H is a halfspace bounded by a hyperplane π, the set C̃ obtained by shaking C on π is defined as the set contained in H, such that for every line ` orthogonal to π, C̃ ∩ ` is a segment of the same length as C ∩ `, and one of its endpoints is on π. It is shown that there exist d + 1 hyperplanes such that every compact set can be reduced to a simplex, via repeated shaking processes on these hyperplanes. MSC 2000: 52A30 Introduction Symmetrizations and more general rearrangements are a powerful tool used to solve many problems in mathematics. We mention for instance their employment in the proof of isoperimetric and functional inequalities (see e.g. [13], [14], [5]). The best known symmetrization is the one introduced by Steiner. Given a compact set C ⊂ R and a hyperplane π, the Steiner symmetral C ′ of C with respect to π is obtained as follows: for every line ` orthogonal to π, C ′ ∩ ` is a segment of the same length as C ∩ `, having its midpoint on π. An essential feature of this process is that every compact set can be reduced to a ball of the same volume via countably many Steiner symmetrizations (see e.g. [2], [7], [8]). The aim of the present paper is to establish an analogous result for another type of rearrangement, namely the shaking process. The set C̃ obtained by shaking C on π is the 1 0138-4821/93 $ 2.50 c ©2001 Heldermann Verlag 124 S. Campi, A. Colesanti, P. Gronchi: Shaking Compact Sets set, contained in one of the halfspaces bounded by π, such that C̃ ∩ ` is again a segment of the same length as C ∩ `, having one endpoint on π. The shaking process (Schüttelung) was introduced by Blaschke in [3] for solving the Sylvester problem in the plane. Blaschke’s argument relies on the fact that, for every line `, the set obtained by shaking a triangle T on ` is affinely equivalent to T . This property characterizes triangles similarly as the affine invariance under Steiner symmetrizations characterizes ellipses. In higher dimensions, while the class of ellipsoids is still closed under Steiner symmetrizations, it is not true that the shaking process maps simplices into simplices. For convex bodies both Steiner and shaking processes can be seen as particular instances of a more general class of transformations which move continuously each chord of a set parallel to a fixed direction. These transformations were used by the authors to approach extremal problems of Sylvester’s type (see [6]). The idea underlying the shaking process can be found in several papers. For instance, Uhrin [15] used it for strengthening the Brunn-Minkowski-Lusternik inequality, and Kleitman [9] introduced a discrete version of this process to obtain discrete isoperimetric inequalities (see also Bollobás and Leader [4]). In [1] Biehl showed that given a convex body K in R there exists a sequence of lines πi, i ∈ N, such that the process of shaking K successively on πi transforms it into a triangle. Furthermore Biehl suggests that his argument can be extended to arbitrary dimension. In fact such an extension is performed by Schöpf in [11]. We notice that in the procedure used by Biehl and Schöpf the choice of πi is recursive, i.e. it depends on the resulting body at the previous step. In the present paper we improve this result in two directions: We extend it to the class of compact sets and we prove that the sequence πi can be chosen independently ofK. More precisely, we consider the simplex S in R whose vertices are at the points (0, 0, . . . , 0), (1, 0, . . . , 0), (0, 1, 0, . . . , 0), . . . , (0, . . . , 0, 1) and an arbitrary compact set C. We show that by shaking repeatedly C on the hyperplanes bounding S we obtain a sequence of sets converging (up to translations) to a simplex homothetic to S, in the Hausdorff metric. The simpler case when C is convex, which will be treated in Section 2, plays a key role in proving the result in its generality (Section 3). We note that since the sequence of hyperplanes can be chosen independently of the compact set, we can transform two compact sets into homothetic simplices simultaneously. This fact can be used to obtain the Brunn-Minkowski-Lusternik inequality (see Remark 3.3). 1. Notations and preliminaries In the d-dimensional Euclidean space R let O denote the origin, e1, e2, . . . , ed the standard orthonormal basis, and R+ = {x ∈ R : xi > 0, i = 1, 2, . . . , d}. Furthermore Sd−1 and B stand for the unit sphere and the unit ball of R, respectively. We denote by C the family of all compact sets in R and by K the subset of C of all convex bodies, i.e. all compact convex sets in R with non-empty interior. Both C and K are endowed with the Hausdorff distance dH and the Minkowski or vector sum (see S. Campi, A. Colesanti, P. Gronchi: Shaking Compact Sets 125 e.g. [10]). If A ∈ C, then |A|d stands for its d-dimensional Lebesgue measure, conv(A), int(A) and ∂A for the convex hull, the interior and the boundary of A, respectively. Let C ∈ C and fix a hyperplane π and a unit vector v orthogonal to π. We define the set Cπ,v obtained by shaking C on π with respect to v as follows. For every x ∈ π, let C(x) be the intersection of C with the straight line through x parallel to v. We define Cπ,v(x) = { ∅ if C(x) = ∅ conv({x, x+ |C(x)|1 v}) if C(x) 6= ∅ and Cπ,v = ⋃