Most Convex Bodies are Isometrically Indivisible

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 C _i 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 $m \in \{2, 3, \ldots, \aleph_0 \}$.