Inner tube formulas for polytopes

TL;DR

This paper proves that the volume of the inner r-neighborhood of a polytope is a piecewise polynomial function of degree d, confirming a conjecture and analyzing its differentiability based on the polytope's geometric properties.

Contribution

It establishes that the inner tube volume function is a pluri-phase Steiner-like polynomial and determines its coefficients for equiangular polytopes, advancing geometric understanding.

Findings

Volume function is a continuous piecewise polynomial of degree d.

Coefficients depend on dihedral angles and skeletal volumes for equiangular polytopes.

Provides bounds and conditions for the function's differentiability.

Abstract

We show that the volume of the inner $r$-neighborhood of a polytope in the $d$-dimensional Euclidean space is a pluri-phase Steiner-like function, i.e. a continuous piecewise polynomial function of degree $d$, proving thus a conjecture of Lapidus and Pearse. In the case when the polytope is dimension-wise equiangular we determine the coefficients of the initial polynomial as functions of the dihedral angles and the skeletal volumes of the polytope. 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 give also sufficient conditions for the highest differentiability degree to be attained.