The authors define the ‘linear scan transform’ G of a set in ℝ’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 integal-geometric formulae used in stereology can be expressed as identifies for G. The authors modify a construction of Waksman (1987) to construct a metric η for ‘regular’ subsets of ℝ’d defined as the L1 distance between their linear scan transforms. For convex sets only, η is topplogically equivalent to the Hausdorff metric. the set covaraince function (of a generally non-convex set) depends continuously on its set argument, with respect to η and the uniform metric on covariance functions.
