My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:
- Families of similar shapes, such as squares, are closed in this metric. Also more general families, such as the family of 2-fat objects, are closed in this metric.
- Converging sequences of interior-disjoint shapes, converge to interior-disjoint limits.
- Every continuous measure on a converging sequence, converges to the measure of the limit.
I tried the Hausdorff distance, which is a metric on the space of closed sets, but found out that it doesn't say much about measures.
I tried the Symmetric distance (defined as the area of the symmetric difference), but found out that it is only a pseudo-metric. I tried to make it a metric by restricting the underlying space of shapes, but found out that it is tricky even when only polygons are considered. I thought of converting the pseudo-metric to a metric on equivalence classes and then selecting a representative shape from each equivalence class, but found no simple way to do this selection.
I thought of defining a new metric which is the maximum of the Hausdorff distance and the Symmetric distance and enjoy the best of the two worlds, but at that point, it began to feel like reinventing the wheel. Surely I am not the first who needs a metric between plain geometric shapes.
So my question is:
Is there a paper or a book that explicitly studies the topic of metrics between shapes in the plane, not in the context of general topology but with attention to the specific geometric properties?