Advances in Geometry
Managing Editor: Grundhöfer, Theo / Joswig, Michael
Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard
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An upper bound on the volume of the symmetric difference of a body and a congruent copy
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.
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