On the measurement of shapes

We are here concerned with the functionf which assigns to each pointP of an object the numberf(P) which is the shortest distance fromP to the border. This function appears in various guises in diverse biological studies. The functionf(P) is itself a measure of shape—or more precisely, an infinite set of measures, one for each point (and hence, in view of its geometric definition, usually in a form inconvenient for use). Thus in this paper we sought a reasonable representative of this infinite set of measures, namely themean of the numbersf(P) asP ranges over all points of the entity. Computability studies are developed for various classes of shapes. For example, (1) the mean for a lamina bounded by a polygon circumscribable about a circle of radiusr isr/3; (2) the mean for a domain bounded by a polyhedron circumscribable about a sphere of radiusr isr/4. The transition from pointwise to piecewisef(P), especially in the non-convex case, requires working with inequalities.