Abstract. We consider maps of the family of convex bodies in Euclidean ddimensional space into itself that are compatible with certain structures on this family: A Minkowski-endomorphism is a continuous, Minkowski-additive map that commutes with rotations. For d ≥ 3, a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by applying the generalized spherical Radon transform to support functions. We give a complete characterization of weakly monotonic Minkowski-endomorphisms. A corresponding theory is developed for Blaschke-endomorphisms, where additivity is now understood with respect to Blaschke-addition. Using a special mixed volume, an adjoining operator can be introduced. This operator allows one to identify the class of Blaschke-endomorphisms with the class of weakly monotonic, non-degenerate and translation-covariant Minkowski-endomorphisms.The following application is also shown: If a (weakly monotonic and) nontrivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball.

We investigate algorithms for reconstructing a convex body K in R n from noisy measurements of its support function or its brightness function in k directions u1, . . . , u k . The key idea of these algorithms is to construct a convex polytope P k whose support function (or brightness function) best approximates the given measurements in the directions u1, . . . , u k (in the least squares sense). The measurement errors are assumed to be stochastically independent and Gaussian.It is shown that this procedure is (strongly) consistent, meaning that, almost surely, P k tends to K in the Hausdorff metric as k → ∞.Here some mild assumptions on the sequence (ui) of directions are needed. Using results from the theory of empirical processes, estimates of rates of convergence are derived, which are first obtained in the L2 metric and then transferred to the Hausdorff metric. Along the way, a new estimate is obtained for the metric entropy of the class of origin-symmetric zonoids contained in the unit ball.Similar results are obtained for the convergence of an algorithm that reconstructs an approximating measure to the directional measure of a stationary fiber process from noisy measurements of its rose of intersections in k directions u1, . . . , u k . Here the Dudley and Prohorov metrics are used. The methods are linked to those employed for the support and brightness function algorithms via the fact that the rose of intersections is the support function of a projection body.While most of the paper is devoted to reconstruction of convex bodies, Section 9 focuses on a problem from stereology, that of reconstructing an unknown directional measure of a stationary fiber process from a finite number of noisy measurements of its rose of intersections. It turns out that the corresponding algorithm, Algorithm NoisyRoseLSQ, is very closely related to Algorithm NoisyBrightLSQ, due to the fact that the rose of intersections is the support function of a projection body. This fact was also used by Kiderlen [20], where an estimation method similar to Algorithm NoisyRoseLSQ was suggested and analyzed. Convergence of Algorithm NoisyRoseLSQ was proved by Männle [23], but also follows easily from our earlier results (see Proposition 9.1). With suitable extra assumptions, we can once again obtain estimates of rates of convergence of the approximating measures to the unknown directional measure. These are first given for the Dudley metric in Theorem 9.4, but can easily be converted to rates for the Prohorov metric. For example, for the Prohorov metric, the rate is of order k −1/20 in the three-dimensional case. 2. Definitions, notation and preliminaries. As usual, S n−1 denotes the unit sphere, B the unit ball, o the origin and · the norm in Euclidean nspace R n . It is assumed throughout that n ≥ 2. A direction is a unit vector, that is, an element of S n−1 . If u is a direction, then u ⊥ is the (n − 1)dimensional subspace orthogonal to u. If x, y ∈ R n , then x · y is the inner product of x and y and [x, y] denotes the line segment with ...

Let B (“black”) and W (“white”) be disjoint compact test sets in ℝd, and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A ⊂ ℝd. If the union B ∪ W is rescaled by a factor tending to zero, then the rescaled volume converges to a value determined by the surface area measure of A and the support functions of B and W, provided that A is regular enough (e.g., polyconvex). An analogous formula is obtained for the case when the conditions B ⊂ A and W ⊂ AC are replaced by prescribed threshold volumes of B in A and W in AC. Applications in stochastic geometry are discussed. First, the hit distribution function of a random set with an arbitrary compact structuring element B is considered. Its derivative at 0 is expressed in terms of the rose of directions and B. An analogous result holds for the hit‐or‐miss function. Second, in a design based setting, different random digitizations of a deterministic set A are treated. It is shown how the number of configurations in such a digitization is related to the surface area measure of A as the lattice distance converges to zero.

a b s t r a c tA local method for estimating surface area and surface area measure of three-dimensional objects from discrete binary images is presented. A weight is assigned to each 2 Â 2 Â 2 configuration of voxels and the total surface area of an object is given by summation of the local area contributions. The method is based on an exact asymptotic result that holds for increasing resolution of the digitization. It states that the number of occurrences of a 2 Â 2 Â 2 configuration is asymptotically proportional to an integral of its ''h-function" with respect to the surface area measure of the object. We find explicit representations for these h-functions. Analyzing them in detail, we determine weights that lead to an asymptotic worst case error for surface area estimation of less than 4%. We show that this worst case error is the best possible. Exploiting the local nature of the asymptotic result, we also establish two parametric estimators for the surface area measure. The latter allow to quantify anisotropy of the object under consideration. Simulation studies illustrate the validity of the estimation procedure also for finite, but sufficiently high resolution.

Abstract. This paper is concerned with various geometric averages of sections or projections of convex bodies. In particular, we consider Minkowski and Blaschke sums of sections as well as Minkowski sums of projections. The main result is a Crofton-type formula for Blaschke sums of sections. This is used to establish connections between the different averages mentioned above. As a consequence, we obtain results which show that, in some circumstances, a convex body is determined by the averages of its sections or projections.

Estimation methods for the directional measure of a stationary planar random set Z, based only on discretized realizations of Z, are discussed. Properties of the discretized set that can be derived by comparing neighbouring grid points are used. Larger grid configurations of more than two grid points are considered. It is shown that the probabilities of observing the various types of configurations can be expressed in terms of the first contact distribution function of Z (with a finite structuring element). An important prerequisite result concerning deterministic dilation areas is also established. The inference on the mean normal measure based on 2×2 configurations is discussed in detail.

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We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. The second algorithm uses harmonic intrinsic volumes which are certain values of the surface tensors and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based on measurements subject to noise is established under certain assumptions on the noise variables.