An upper bound on the volume of the symmetric difference of a body and a congruent copy

Abstract

Abstract Let A be a bounded subset of ℝd for some d ≥ 2. We give an upper bound on the volume of the symmetric difference of A and ƒ(A) where f is a translation, a rotation, or the composition of both, a rigid motion. We bound the volume of the symmetric difference of A and f(A) in terms of the (d - 1)- dimensional volume of the boundary of A and the maximal distance of a boundary point to its image under ƒ. The boundary is measured by the (d - 1)-dimensional Hausdorff measure, which matches the surface area for sufficiently nice sets. In the case of translations, our bound is sharp. In the case of rotations, we get a sharp bound under the assumption that the boundary is sufficiently nice. The motivation to study these bounds comes from shape matching.