Abstract
The classical Steiner formula states that for a convex body \(K\in \mathcal {K}^n\) the volume Vn(K + ϱBn) of the parallel set K + ϱBn, for ϱ ≥ 0, is a polynomial in ϱ ≥ 0 whose coefficients (if suitably normalized) are the intrinsic volumes (or Minkowski functionals) V0(K), …, Vn(K) evaluated at the given convex body K. Among these functionals are the volume Vn and the surface area Vn−1 (up to a constant factor), but also less known functionals such as the mean width (which is proportional to V1) and the Euler characteristic V0. The intrinsic volumes are continuous, additive and motion invariant functionals on convex bodies. In Chap. 5 we shall see that these properties characterize the intrinsic volumes. In addition, the intrinsic volumes are subject to various inequalities, of which the celebrated isoperimetric inequality is a particular example.
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