Background from Convex Geometry

Abstract

Let K be a convex body (i.e., a nonempty, convex and compact set) in $\mathbb {R}^d$. We use the notation for the r-parallel body (r ≥ 0)

$$\displaystyle K_r:=\{ z\in \mathbb {R}^d :\, \mathrm {dist}\, (z,K)\leq r\}. $$

(Equivalently, we can write using Minkowski summation K _r = K + rB ^d, where B ^d is the unit ball centred in the origin in $\mathbb {R}^d$.) The Steiner formula expresses the volume of K _r as a polynomial:

$$\displaystyle \mathcal {L}^d(K_r)=\sum _{k=0}^d\omega _kr^kV_{d-k}(K),\quad r\geq 0. $$

The coefficient V _k(K) is called the k-th intrinsic volume of K (k = 0, 1, …, d).