A stereological metric for plane domains (Q1099469)

In this paper we consider the chord length distribution \(\omega_ S(t)\) for a plane domain S [see \textit{R. Sulanke}, Math. Nachr. 23, 51-74 (1961; Zbl 0101.404)]. Sulanke has shown that when s is convex \(\omega_ S(t)\) is a continuous function of t, the chord length. We generalize this to non-convex S with a restriction on the differentiability of the boundary of S. The main goal of this paper is not, however, to prove continuity in t, but rather in S. Thus we define a metric \(\gamma\) (\(\cdot,\cdot)\) on plane domains such that \(\gamma (S_ n,S)\to 0\) guarantees that \(\omega_{S_ n}\) converges uniformly to \(\omega_ S.\) We also consider a function \(B_ S(t)\) equivalent to the associated function of S and prove the analogous results for \(B_ S(t)\); in this case we show that \(S\to B_ S\) is a Lipschitz continuous mapping from a metric space of domains into \(L^ 1[0,D]\) where D is the diameter of a large disc containing the domains. The main method of the paper is the analysis of glance functions which describe how a line meets a domain, they relate such quantities as the number of components of the intersection, the sum of the lengths of the components, and the diameter of the intersection. These were studied by the author in Trans. Am. Math. Soc. 296, 507-520 (1986; Zbl 0594.62018). In the conclusion we make some remarks about how these ideas could be generalized to \(R^ n\) for \(n>2\).