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Proceedings of the American Mathematical Society

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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

Andrei V. Ratiu
Affiliation: Department of Mathematics and Statistics, University of Melbourne, Parkville, Melbourne, VIC 3010, Australia

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.