A new trick for for Santalo's integral geometric proof of the strong Bonnesen inequality

Given a lamina $K$ whose boundary $\partial K$ is convex we define the Bonnesen functional by integrating over the position and orientation of a disk of radius $r$ its intersections with the lamina and its boundary. $$B(r)=\frac{1}{2\pi }\int (\frac{n}{2}-\nu )dxdyd\theta =rL-A-\pi r^2$$ where $n$ is the number of intersections of the boundary of $K$ with the boundary of the disk and $\nu$ (with values either 0 or 1) is the number of intersections of the interiors. Analyzing the interval on which $B(r)\ge 0$ leads to a number of isoperimetric inequalities. For example Santalo observed that $B(r)$ is non-negative on $[r_{in},r_{out}]$ where $r_{in}$, the inradius, is the largest disk contained in $K$ and $r_{out}$, the outradius, is the smallest disk containing $K$. For this configuration if the bodies intersect than their boundaries must intersect in 2 or more points (generically) so the integrand is always non-negative and $B(r)\ge 0$. In this paper we show that $B(r)\ge 0$ on the interval$[{\rho }_{in},{\rho }_{out}]$ where these are the inner and outer radii of the annulus of minimal width which surrounds $\partial K$. In this case the integrand is not always positive but we carefully analyze the integral and use an "averaging trick" to balance regions where the integrand is negative with regions where the integrand is positive to show that $B(r)\ge 0$ The final isoperimetric inequalities obtained are not new, the same results have been obtained by cleverly cutting $\partial K$ and doubling it to create two centrally symmetric curves and analyzing those, but I believe that the averaging trick is new in this context. The paper also reviews the concept of "positive centers" and shows how the isoperimetric inequalities can be used to the show that the final shape of the curve shortening flow in Euclidean and Minkowski geometries are respectively the disk and the isoperimetrix.