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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Inner tube formulas for polytopes


Authors: Şahin Koçak and Andrei V. Ratiu
Journal: Proc. Amer. Math. Soc. 140 (2012), 999-1010
MSC (2010): Primary 52B11; Secondary 52A38
Published electronically: September 1, 2011
MathSciNet review: 2869084
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Abstract: We show that the volume of the inner $ r$-neighborhood of a polytope in the $ d$-dimensional Euclidean space is a pluriphase Steiner-like function, i.e. a continuous piecewise polynomial function of degree $ d$, thus proving a conjecture of Lapidus and Pearse. We discuss also the degree of differentiability of this function and give a lower bound in terms of the set of normal vectors of the hyperplanes defining the polytope. We also give sufficient conditions for the highest differentiability degree to be attained.


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Additional Information

Şahin Koçak
Affiliation: Department of Mathematics, Anadolu University, 26470 Eskişehir, Turkey
Email: skocak@anadolu.edu.tr

Andrei V. Ratiu
Affiliation: Department of Mathematics and Statistics, University of Melbourne, Parkville, Melbourne, VIC 3010, Australia
Email: aratiu@unimelb.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11307-8
PII: S 0002-9939(2011)11307-8
Keywords: Polytope, Steiner formula, tube formula
Received by editor(s): December 8, 2010
Published electronically: September 1, 2011
Communicated by: Ken Ono
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.