  Applications such as political redistricting demand quantitative measures of
geometric compactness to distinguish between simple and contorted shapes. While
the isoperimetric quotient, or ratio of area to perimeter squared, is commonly
used in practice, it is sensitive to noisy data and irrelevant geographic
features like coastline. These issues are addressed in theory by the
isoperimetric profile, which plots the minimum perimeter needed to inscribe
regions of different prescribed areas within the boundary of a shape. Efficient
algorithms for computing this profile, however, are not known in practice.
Hence, in this paper, we propose a convex Eulerian relaxation of the
isoperimetric profile using total variation. We prove theoretical properties of
our relaxation, showing that it still satisfies an isoperimetric inequality and
yields a convex function of the prescribed area. Furthermore, we provide a
discretization of the problem, an optimization technique, and experiments
demonstrating the value of our relaxation.
