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For a convex set in Euclidean space, there is an obvious notion of its center: namely, its center of mass, which by convexity lies in the set. For a nonconvex set there is just as obviously no nice notion of its center, even if we restrict consideration to a connected component. (As a tangential aside, recall the balancing bird toy.)

To narrow things a bit, let's consider a connected (but not necessarily topologically trivial) region in the plane with smooth boundary. One notion of central points of this region is furnished by taking erosions by disks of varying radius; another is furnished by considering the heat equation with initial data 1 on the boundary and 0 in the interior, then looking for points of minimal temperature (it is not quite clear to me that these points are fixed over time, but the alternative seems very unlikely).

I am curious to know (if and) where such notions have been studied, and ideally surveyed/catalogued. Higher dimensions would be nice to know about too.

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  • $\begingroup$ How would you find the central point of an annulus? $\endgroup$
    – Ilya Bogdanov
    Commented Apr 18, 2016 at 15:50
  • $\begingroup$ @IlyaBogdanov- Either of the two methods I propose (eroding from the boundary or using the heat equation) would isolate a central circle lying entirely within the annulus. $\endgroup$
    – Steve Huntsman
    Commented Apr 18, 2016 at 15:56
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    $\begingroup$ In the heat equation, I'm assuming you mean the initial condition is $0$ but the boundary condition is $1$. But except in special cases it seems to me very unlikely that the points of minimum temperature would be independent of time. The solution will be of the form $u(x,t) = 1 + \sum_{j} a_j \exp(-\lambda_j t) u_j(x)$ where $u_j$, $\lambda_j$ are eigenfunctions and eigenvalues of the Laplacian on your region. Why should the minima of this not be dependent on $t$? $\endgroup$
    – Robert Israel
    Commented Apr 18, 2016 at 22:04
  • $\begingroup$ @RobertIsrael- My casual physical intuition led me to (essentially) expect the minima to be always determined by the most rapidly decaying eigencomponent. But I certainly doubt that now in light of your comment. On the other hand, perhaps a short-time limit would work here. $\endgroup$
    – Steve Huntsman
    Commented Apr 19, 2016 at 0:55

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In a recent paper,

Centers of disks in Riemannian manifolds,

Igor Belegradek and I study whether it is possible to extend to nonconvex objects the notion of center of mass or other classical centers associated to convex bodies. We prove the existence of a center, or continuous selection of a point, in the relative interior of C1 embedded k-disks in Riemannian n-manifolds. If k ≤ 3 the center can be made equivariant with respect to the isometries of the manifold, and under mild assumptions the same holds for k = 4 = n . But for every n ≥ k ≥ 6 there are examples where an equivariant center does not exist. The center can be chosen to agree with any of the classical centers (e.g., center of mass, circumcenter, Steiner point) defined on the set of convex bodies in the Euclidean space.

Addendum [August 14, 2023]: In a more recent paper,

Point selections from Jordan domains in Riemannian surfaces,

Igor Belegradek and I use conformal maps, specifically Caratheodory's kernel theorem, to extend the results of the previous paper to topological domains in dimension $2$, as discussed also in this post.

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