Line transects, covariance functions and set convergence

We define the .linear scan transform' G of a set in .d using information observable on its one-dimensional linear transects. This transform determines the set covariance function, interpoint distance distribution, and (for convex sets) the chord length distribution. Many basic integral-geometric formulae used in stereology can be expressed as identities for G. We modify a construction of Waksman (1987) to construct a metric . for .regular' subsets of .d defined as the L1 distance between their linear scan transforms. For convex sets only, . is topologically equivalent to the Hausdorff metric. The set covariance function (of a generally non-convex set) depends continuously on its set argument, with respect to . and the uniform metric on covariance functions.