Chapter 4 Nonasymptotic densities 1 4.1 Introduction 78 $^{1}$Previously published as [27] This work discusses the integral area invariant introduced by Manay et al. [30], par- ticularly with regard to reconstructability of shapes. This topic has been considered previously by Fidler et al. [20] [21] for the case of star-shaped regions. Recent results have shown local injectivity in the neighborhood of a circle [6] and for graphs in a neighborhood of constant functions [8]. The present work does not assume a star-shaped condition but does make use of a tangent-cone graph-like condition which is local to the integral area circle. We also present an interpretation of the integral area invariant as a nonasymptotic density. This is based on a poster presented by the authors [26]. Our tangentially graph-like and tangent-cone graph-like conditions (definitions 4.2.3 and 4.2.5 in section 4.2) restrict our attention to shapes with boundaries that can lo- cally (i.e., within radius r ) be viewed as graphs of functions in a Cartesian plane in one particular orientation (in the case of tangentially graph-like) or a particular set of ori- entations (for tangent-cone graph-like). Intuitively, these conditions guarantee that the boundary does not turn too sharply within the given radius and that working locally in Euclidean space is the same as working locally on the boundary of our shapes (i.e., the shape boundary does not pass through any given invariant circle multiple times, section