Inner tube formulas for polytopes

Koçak, Şahin; Ratiu, Andrei

doi:10.1090/S0002-9939-2011-11307-8

Inner tube formulas for polytopes
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by Şahin Koçak and Andrei V. Ratiu PDF

Proc. Amer. Math. Soc. 140 (2012), 999-1010 Request permission

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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