Remark 4.2.2. Note the lack of the normalizing factor πr 2 in the definition of g ( s, r ) . Since we presume that r is fixed and known for the situations we study, it’s trivial to convert data between the forms g ( s, r ) and g ( s,r ) πr 2 ; we choose to leave out the normalizing factor in the definition of g ( s, r ) as it is the integral area invariant of Manay et al. [30] and this form proves useful when computing derivatives in section 4.4. We introduce the tangentially graph-like condition as a simplifying assumption for the shapes we consider. Definition 4.2.3. For a fixed radius r , we say that ∂ Ω is graph-like (GL) at a point p ∈ ∂ Ω (or graph-like on D ( p, r ) ) if it is possible to impose a Cartesian coordinate system such that the set of points ∂ Ω ∩ D ( p, r ) is the graph of some function f in this coordinate system. Without loss of generality, we adopt the convention that p is the origin so that f (0) = 0 . We define tangentially graph-like (TGL) in the same way but further require that ∂ Ω be continuously differentiable and f $^{′}$(0) = 0 (noting that f is C 1 because ∂ Ω is). This is illustrated in figure 4.2(a). Without loss of generality (and in keeping with our convention that γ traverses ∂ Ω counterclockwise), we assume that the interior of Ω is “up” in the circle (i.e., that (0 , ϵ ) ∈ Ω for sufficiently small ϵ > 0 ). If ∂ Ω is (tangentially) graph-like on D ( p, r ) for all p ∈ ∂ Ω , we say that ∂ Ω is (tangentially) graph-like for radius r . It is instructive to consider what is not graph-like or tangentially graph-like. Viola- tions of the graph-like condition are generally due to a radius that is too large (certainly, choosing a radius so large that all of Ω is in the disk will do it). For example, a unit side length square is not graph-like with radius 1 2 + ϵ for any ϵ > 0 (position the circle at the center of a side; see figure 4.3(a)). Notice that the same square is graph-like with any radius 1 2 or below. A shape can fail to be tangentially graph-like while still being 83