The derivative of the parallel volume difference (Q2867269)

The paper under review is devoted to the study of the asymptotic behavior of the parallel volume in Minkowski spaces as the distance tends to infinity. One of the main results (Theorem~1.1) is the following: For any body \(K \subset \mathbb{R}^2\), the function \(r \mapsto V_2(\mathrm{conv}(K)+rB^2)-V_2(K+rB^2)\) (where \(B^2\) is the \(2\)-dimensional closed unit ball, \(\mathrm{conv}(K)\) is the convex hull of \(K\), and \(V_2\) is the 2-dimensional Lebesgue measure), is differentiable for almost all sufficient large \(r\) and there is a constant \(c\) such that \(|\frac{d}{dr}(V_2(K+rB^2)-V_2(\mathrm{conv}(K)+rB^2))|<c\cdot r^{-2}\) for all large enough \(r\) in which derivative exists. This result is generalized for random bodies in \(\mathbb{R}^n\) (Theorem~1.2). As applications of the above results, the author presents an asymptotic formula for the boundary length of a planar Wiener sausage and examines contact distributions of Boolean models.