Analysis of Incomplete Data and an IntrinsicDimension Helly Theorem The analysis of incomplete data is a longstanding challenge in practical statistics. When, as is typical, data objects are represented by points in xe2x84x9d d , incomplete data objects correspond to affine subspaces (lines or xcex94flats). With this motivation we study the problem of finding the minimum intersection radius r(xe2x84x92) of a set of lines or xcex94flats xe2x84x92: the least r such that there is a ball of radius r intersecting every flat inxc2xa0xe2x84x92. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higherdimensional flats, primarily because xe2x80x9cdistancesxe2x80x9d between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Hellyxe2x80x99s theorem. This xe2x80x9cintrinsicdimensionxe2x80x9d Helly theorem states: for any family xe2x84x92 of xcex94dimensional convex sets in a Hilbert space, there exist xcex942 sets xe2x84x92xe2x80xb2xe2x8ax86xe2x84x92 such that r(xe2x84x92)xe2x89xa42r(xe2x84x92xe2x80xb2). Based upon this we present an algorithm that computes a (1e)core set xe2x84x92xe2x80xb2xe2x8ax86xe2x84x92, xe2x84x92xe2x80xb2O(xcex94 4e), such that the ball centered at a point c with radius (1e)r(xe2x84x92xe2x80xb2) intersects every element ofxc2xa0xe2x84x92. The running time of the algorithm is O(n xcex941 dpolyxe2x80x89(xcex94e)). For the case of lines or line segments (xcex941), the (expected) running time of the algorithm can be improved to O(ndpolyxe2x80x89(1e)). We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient  to and :255,handled by Pankaj Agarwal.

Uniqueness of the measurement function in Croftons formula Abstract   Croftonu0027s intersection formula states that the    (  n  xe2x88x92  j  )   th intrinsic volume of a compact convex set in      R    n      can be obtained as an invariant integral of the    (  k  xe2x88x92  j  )   th intrinsic volume of sections with kplanes. This paper discusses the question if the    (  k  xe2x88x92  j  )   th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Croftonu0027s formula is unique.  The answer is negative: we show that the sums of the    (  k  xe2x88x92  j  )   th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree k yield counterexamples. If the measurement function is local, these turn out to be the only examples when    k    1    or when    k    2    and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when    k  xe2x89xa5  2   .