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

Abstract

Let A be a bounded subset of IR^d. We give an upper bound on the volume of the symmetric difference of A and f(A) where f is a translation, a rotation, or the composition of both, a rigid motion. The volume is measured by the d-dimensional Hausdorff measure, which coincides with the Lebesgue measure for Lebesgue measurable sets. 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 f. 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. For two shapes A and B in IR^d and a class of transformations, the matching problem asks for a transformation f such that f(A) and B match optimally. The quality of the match is measured by some similarity measure, for instance the volume of overlap. Let A and B be bounded subsets of IR^d, and let F be the function that maps a rigid motion r to the volume of overlap of r(A) and B. Maximizing this function is a shape matching problem, and knowing that F is Lipschitz continuous helps to solve it. We apply our results to bound the difference |F(r) - F(s)| for rigid motions r,s that are close, implying that F is Lipschitz continuous for many metrics on the space of rigid motions. Depending on the metric, also a Lipschitz constant can be deduced from the bound.