Some properties of chord length distributions

The chord length distributions of planar convex sets are discussed, particu- larly the density values at the extremes of the range; there is a qualitative distinction between polygons and sets with smooth boundaries. The distance between convex sets is related to the distance between distribution functions. This paper is concerned with recording some of the simpler properties of chord length distributions (henceforth abbreviated to c.l.d.) of compact con- vex figures in R2. Detailed calculation of c.l.d.'s are exercises in integration. In the case of polygons the results can be expressed in terms of elementary functions, but are usually highly tedious, requiring different expressions over different ranges. In this paper Kwill denote a compact convex set in R2, containing more than one point; Jr will denote the set of such sets and ((.) the set of equivalence classes of.J under the Euclidean group of motions, M2. The measure on the set of lines in R2 will be the M2 invariant measure, u. If a line G meets K we may write G t K and the set of incident lines as G(K), with S(G) the chord length. If an origin is chosen in K a line can be parametrised by its perpendicular distance, p, and the orientation p of that perpendicular. It is well known (Santal6 (13), p. 28) that the invariant line measure is dG = dpd4, so that, in this parametrisation, ,L is identified with Lebesgue measure, u2, in R2. It is shown in Sulanke (14) that S is a measurable function on G(K). As #u(G(K)) is equal to the perimeter, L, of K the distribution function of chord length is