Jump to ContentJump to Main Navigation
Show Summary Details
More options …

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

4 Issues per year


IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

Online
ISSN
1615-7168
See all formats and pricing
More options …
Volume 14, Issue 2

Issues

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

D. Schymura
Published Online: 2014-04-01 | DOI: https://doi.org/10.1515/advgeom-2013-0029

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.

About the article

Published Online: 2014-04-01

Published in Print: 2014-03-01


Citation Information: Advances in Geometry, Volume 14, Issue 2, Pages 287–298, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2013-0029.

Export Citation

©2014 by Walter de Gruyter Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
M. Milman and J. Xiao
Mathematische Nachrichten, 2017
[2]
Matthew M. Dunlop and Andrew M. Stuart
Inverse Problems and Imaging, 2016, Volume 10, Number 4, Page 1007
[3]
Matteo Cozzi
Annali di Matematica Pura ed Applicata (1923 -), 2017, Volume 196, Number 2, Page 555

Comments (0)

Please log in or register to comment.
Log in