Putnam 1976 A5

A-5. In the $(x, y)$-plane, if $R$ is the set of points inside and on a convex polygon, let $D(x, y)$ be the distance from $(x, y)$ to the nearest point of $R$. (a) Show that there exist constants $a, b$, and $c$, independent of $R$, such that $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-D(x, y)} d x d y=a+b L+c A$$ where $L$ is the perimeter of $R$ and $A$ is the area of $R$. (b) Find the values of $a, b$, and $c$.