Jürgen Kampf

On the convex hull of symmetric stable processes

Let α ∈ (1, 2] and X be an Rd-valued α-stable process with independent and symmetric components starting in 0. We consider the closure St of the path described by X on the interval [0, t] and its convex hull Zt. The first result of this paper provides a formula for certain mean mixed volumes of Zt and in particular for the expected first intrinsic volume of Zt. The second result deals with the asymptotics of the expected volume of the stable sausage Zt+B (where B is an arbitrary convex body with interior points) as t→ 0.

A limitation of the estimation of intrinsic volumes via pixel configuration counts

It is often helpful to compute the intrinsic volumes of a set of which only a pixel image is observed. A computational efficient approach, which is suggested by several authors and used in practice, is to approximate the intrinsic volumes by a linear functional of the pixel configuration histogram. Here we want to examine, whether there is an optimal way of choosing this linear functional, where we will use a quite natural optimality criterion that has already been applied successfully for the estimation of the surface area. We will see that for intrinsic volumes other than volume or surface area this optimality criterion cannot be used, since estimators which ignore the data and return constant values are optimal w.r.t. this criterion. This shows that one has to be very careful, when intrinsic volumes are approximated by a linear functional of the pixel configuration histogram.