Inner tube formulas for polytopes (Q2880661)

Motivated by the well-known Steiner formula for the volume of extrinsic \(r\)-neighborhoods of convex compact sets in \(\mathbb{R}^d\), the authors are interested in its intrinsic counterparts. Namely, let \(P\) be a convex compact polytope in \(\mathbb{R}^d\). Given \(r>0\), the \(r\)-interior of \(P\) is defined as \(P(r) = \{ Q\in P\mid d(Q,\partial P) \geq r\}\), whereas \(P\setminus P(r)\) is called the inner \(r\)-neighborhood of \(\partial P\). What can one say about the volume of \(P(r)\) viewed as a function of \(r\geq 0\)? The main result states that the volume \(V(r)\) is a continuous piecewise polynomial function. This proves Conjecture 1 formulated in [\textit{M. L. Lapidus} and \textit{E. Pearse}, Proc. Symp. Pure Math. 77, 211--230 (2008; Zbl 1157.28001)]. Moreover, the degree of differentiability of the volume function \(V(r)\) is discussed, and sufficient conditions for the highest differentiability degree \(d-1\) to be attained are proved.