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

作者：Schymura Daria^*

来源：Advances in Geometry, 2014, 14(2): 287-298.

DOI：10.1515/advgeom-2013-0029

Let A be a bounded subset of R-d for some d %26gt;= 2. 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. %26lt;br%26gt;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. %26lt;br%26gt;The motivation to study these bounds comes from shape matching.

• 出版日期2014-3