IntroductionThe theory of P m distributions of stationary point processes and random m w u r e s in Rd, or more generally, on locally compact ABELean groups has d8entially been stimulated by the work of KEBSTAN, Ma== and MEOEE (see, e.g. [8], [el). In one of the classical papers (RYLLNUDZEWSKX [ll]) concerning this topic an interpretation as conditional distributions given a point is worked out which does not use the invariance structure (cf. also JAQERS [6], P a p l~~a l c~o~ [lo]). A systematic treatment and extensions of the laat approach may be found in KALLENBEBO [6].In the meantime these concepts and methods have been applied in stochastic geometry to random plane procasea, random tessellations, and to more general random sets and measures, in particular, in the fractal w e . Most of the related stochastic models deal with invariant structures in euclidean space. Since corresponding integralgeometric results are known in spaces of constant curvature and in more general homogeneous RrEMA"ian spaces it appears appropriate to extend the P u theory to arbitrary homogeneous spaces. Here the invariance properties h e by the associated transformation group which needs not be unimodular and whose dimension is, in general, larger than that of the underlying space. Therefore we obtain some differences to the euclidean caae. Nevertheless, the basic relations remain valid in a modified version. First results are presented in the paper. (In the case when the group does not act transitively, the decomposition given in KRIUKEBERQ [7] might be fruitful.) 1.Notations Let X be a locally compad Polish apace and let 3E be the a-algebra of BOREL sets in X We denote by M the set of all locally finite measures on [X, 33 and by ' $ I the smallest o-algebra on M making all functions p --f p(B) measurable, y E M , B E 2. Furthermore, for any measurable space [ Y, 91, %( Y) is the space of real valued nonnegative '@meesurable functions on Y.

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The nonlinear eigenvalue problem for p-Laplacianis considered. We assume that 1 < p < N and that the functionfis of subcritical growth with respect to the variable u. The existence and C'."-regularity of the weak solution is proved. 9*

The absolute curvature measures for sets of positive reach in R d introduced in [7] satisfy the following kinematic relations: Their integrated values on the intersections with (or on the tangential projections onto) uniformly moved p-planes are constant multiples of the corresponding absolute curvature measures of the primary set. In the special case of convex bodies the first result is the so-called Crofton formula. An analogue for signed curvature measures is well known in the differential geometry of smooth manifolds, but the motion of absolute curvatures used there does not lead to this property. For the special case of smooth compact hypermanifolds our absolute curvature measures agree with those introduced by Santal6 [4] with other methods.In the appendix, the section formula is applied to motion invariant random sets. EQUIVALENT DEFINITIONSThe present paper is a continuation of [7], which also contains motivations and references to the related literature. We first recall some notions introduced there. At the same time we correct an error in formulas (3)-(6) of [7], where certain projection Jacobians have to be inserted. In the sequel, X denotes an arbitrary subset of R d of positive reach (in the sense of Federer [1]) and B a bounded Borel set in R d x S d-1 (or in Rd). Note that convex bodies and compact C2-submanifolds are special cases for X. In [7] the kth absolute curvature measure c~bs(x, B), k = 0 ..... d-1, is introduced by means of the natural invariant measure of the set of all (d-1 -k)-planes locally colliding with X inside B. Thereby affine and linear subspaces of R d are considered as dements of sufficiently large Euclidean spaces via representation by multivectors. Let nor X be the unit normal bundle of X and G(X, d, k) its kth Grassmann bundle (denoted in [7] by G(X, k)), i.e. G(X, d, k)= {(x, n, V): (x, n)~nor X, V~G,(d-1, k)} where G,(d-1, k) is the Grassmann submanifold of G(d, k) of those kdimensional linear subspaces of R d which are orthogonal to the unit vector n ES d-1. Let v" be the normalized rotation invariant measure on d-l,k G,(d-l,k). 17 v is the orthogonal projection onto the subspace V The function f(x, n, V) = (IIv~ x, V) maps G(X, d, k) onto the set sO(X, d, k) of parametrized k-dimensional affine subspaces (k-planes) locally colliding with X. Let P be the coordinate Geometriae Dedicata 41: 229-240, 1992.

The existence of positive solutions, vanishing at infinity, for the semilinear eigenvalue problem Lu = X • f(x, y) in R^ is obtained, where L is a strictly elliptic operator. The function / is assumed to be of subcritical growth with respect to the variable u .