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

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.

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In this paper we continue the investigation of the asymptotic behavior of the parallel volume in Minkowski spaces as the distance tends to infinity that was started in [8] and [9]. Our main result is that the derivative of the difference between the parallel volume of the convex hull of a planar body and the parallel volume of the body itself tends to 0 for r → ∞ at order r −2 . We will use this result to examine Brownian paths and Boolean models.

In this paper we examine the asymptotic behavior of the parallel volume of planar non-convex bodies as the distance tends to infinity. We show that the difference between the parallel volume of the convex hull of a body and the parallel volume of the body itself tends to 0. This yields a new proof for the fact that a planar body can only have polynomial parallel volume, if it is convex. Extensions to Minkowski spaces and random sets are also discussed.