Integral geometry for the 1-norm

Leinster, T. (2012) Integral geometry for the 1-norm. Advances in Applied Mathematics, 49(2), pp. 81-96. (doi: 10.1016/j.aam.2012.02.002)

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Publisher's URL: http://dx.doi.org/10.1016/j.aam.2012.02.002

Abstract

Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in R<sup>n</sup> equipped with the metric derived from the p -norm. This has, in effect, been investigated intensively for 1<p<∞, but not for p=1. We show that integral geometry for the 1-norm bears a striking resemblance to integral geometry for the 2-norm, but is radically different from that for all other values of p . We prove a Hadwiger-type theorem for R<sup>n</sup> with the 1-norm, and analogues of the classical formulas of Steiner, Crofton and Kubota. We also prove principal and higher kinematic formulas. Each of these results is closely analogous to its Euclidean counterpart, yet the proofs are quite different.<p></p>

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Leinster, Dr Tom
Authors: Leinster, T.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Advances in Applied Mathematics
Publisher:Elsevier
ISSN:0196-8858
ISSN (Online):1090-2074
Published Online:22 March 2012
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
424771Self similarity - recursively definable objects in topology, analysis category theory and algebraThomas LeinsterEngineering & Physical Sciences Research Council (EPSRC)EP/D073537/1Mathematics