In this paper we generalize some of the classical results of Sulanke (1963), (1964) in the context of spindle convexity. A planar convex disc S is spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.

General L p dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn-Minkowski theory and the dual Brunn-Minkowski theory. L p dual curvature measures arise from qth dual inrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang [24] formulated the L p dual Minkowski problem, which concerns the characterization of L p dual curvature measures.In this paper, we solve the existence part of the L p dual Minkowski problem for p > 1 and q > 0, and we also discuss the regularity of the solution.

Let K be a d dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by Kn the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the s-th intrinsic volumes Vs(Kn) of Kn for s ∈ {1, . . . , d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the Economic Cap Covering Theorem of Bárány and Larman, and the Efron-Stein jackknife inequality. NotationWe shall work in d-dimensional Euclidean space R d , with origin o, and scalar product ·, · , and induced norm · . The dimension d will be fixed throughout the paper. We shall not distinguish between the Euclidean space and the underlying vector space, and we will use the words point and vector interchangeably, as we need them. Points of R d are denoted by small-case letters of the roman alphabet, and sets by capitals. For reals we use either Greek letters or small-case letters. B j stands for the j-dimensional ball of radius 1 centered at the origin, S j−1 denotes the boundary of B j and κ j denotes the volume of B j . Note that any point x ∈ ∂B j = S j−1 can be considered as a point of the boundary of B j and also as an outer normal to B j at the point x. For a point set T ⊂ R d , we denote the convex hull of T by conv T or simply by [T ]. A compact convex set K with nonempty interior is called a convex body.The intrinsic volumes V s (K), s = 0, . . . , d of a convex body K can be introduced as coefficients of the Steiner formulawhere K + λB d is the Minkowski sum of K and λB d of radius λ ≥ 0. In particular, V d is the volume functional, V 0 (K) = 1, V 1 is proportional to the *