Most Convex Bodies are Isometrically Indivisible

TLDR: A set C in the Euclidean space is called isometrically m-divisible if there exists a disjoint decomposition of C into m subsets Ci pairwise congruent with respect to the group of isometries as discussed by the authors.

Abstract: A set C in the Euclidean space \({\mathbb{R}}^d\) is called isometrically m-divisible if there exists a disjoint decomposition of C into m subsets Ci pairwise congruent with respect to the group of Euclidean isometries. We present a necessary condition for the isometric m-divisibility, which shows in particular that most convex bodies – in the sense of Baire category – are not isometrically m-divisible for any choice of m\(m \in \{2, 3, \ldots, \aleph_0 \}\).