A metric space of geometric shapes My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:


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*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?
 A: The following paper gives an overview on Riemannian geometries on shape spaces and diffeomorphism group.


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*Martin Bauer, Martins Bruveris, Peter W. Michor: Overview of the Geometries of Shape Spaces and Diffeomorphism Groups. Journal of Mathematical Imaging and Vision, 50, 1-2, 60-97, 2014. (pdf)
Edit:
A metric on the space of plane shapes that it somewhat more easy to use (since it allows for explicit solutions of the PDE which is the geodesic equation, but it does not see translations) is in the following paper:


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*Martin Bauer, Martins Bruveris, Stephen Marsland, Peter W. Michor: Constructing reparametrization invariant metrics on spaces of plane curves. Differential Geometry and its Applications 34 (2014), 139–165. (pdf)
A: I can't say for certain if this satisfies all of your requirements, but this paper of Sharon and Mumford studies 2D shapes by comparing the (suitably normalized) Riemann maps from its interior and exterior to the unit circle, by comparing these maps on the circle one represents the shape by an equivalence class of diffeomorphisms of the circle, and these can be made into a metric space. (They require smooth shapes; I don't know if this can be weakened.)
A: Your symmetric distance is fine, as long as we're particularly careful in the definition of a 'shape'. I believe the following will suffice:
A shape is an equivalence class of Borel sets, where two Borel sets are equivalent if their symmetric difference has zero Lebesgue measure.
Then this is a true metric, by definition, and it's possible to take limits of Cauchy sequences: given a Cauchy sequence $A_1, A_2, \dots$ of Borel sets, we define $A_{\omega}$ to consist of all points which occur in cofinitely many $A_i$, no points which occur in finitely many $A_i$, and we don't care about any other points (since their total Lebesgue measure is zero). And it's easy to show that such a limit can be constructed by careful application of countable unions and intersections, so is indeed Borel.
A: You may check papers by Stefan Banach on measure theory. He would consider metric invariant finitely additive measures but for arbitrary sets--in $\mathbb R^2$ too.
Also, certain ideas by Karol Borsuk, in a paper presented at the Mathematical Congress of 1954 (I am quite sure it was 1954) may be of interest to you. Karol Borsuk considered several metrics for compact spaces which were sensitive to topological properties.
BTW, in my opinion the condition on limit being disjoint for the disjoint shapes seems unnatural to me.
