Continuity and diameters of connected sets

The definition of uniform continuity (if it’s done right) can be phrased as: {f\colon X\to Y} is uniformly continuous if there exists a function {\omega\colon (0,\infty)\to (0,\infty)}, with {\omega(0+)=0}, such that {\textrm{diam}\, f(E)\le \omega (\textrm{diam}\, E)} for every set {E\subset X}. Indeed, when {E} is a two-point set {\{a,b\}} this is the same as {|f(a)-f(b)|\le \omega(|a-b|)}, the modulus of continuity. Allowing general sets {E} does not change anything, since the diameter is determined by two-point subsets.

Does it make a difference if we ask for {\textrm{diam}\, f(E)\le \omega (\textrm{diam}\, E)} only for connected sets {E}? For functions defined on the real line, or on an interval of the line, there is no difference: we can just consider the intervals {[a,b]} and obtain

{|f(a)-f(b)|\le \textrm{diam}\, f([a,b]) \le \omega(|a-b|)}

as before.

However, the situation does change for maps defined on a non-convex domain. Consider the principal branch of square root, {f(z)=\sqrt{z}}, defined on the slit plane {G=\mathbb C\setminus (-\infty, 0]}.

Conformal map of a slit domain is not uniformly continuous

This function is continuous on {G} but not uniformly continuous, since {f(-1 \pm i y) \to \pm i } as {y\to 0+}. Yet, it satisfies {\textrm{diam}\, f(E)\le \omega(\textrm{diam}\, E)} for connected subsets {E\subset G}, where one can take {\omega(\delta)=2\sqrt{\delta}}. I won’t do the estimates; let’s just note that although the points {-1 \pm i y} are close to each other, any connected subset of {G} containing both of them has diameter greater than 1.

These points are far apart with respect to the inner diameter metric

In a way, this is still uniform continuity, just with respect to a different metric. Given a metric space {(X,d)}, one can define inner diameter metric {\rho} on {X} by letting {\rho(a,b)} be the infimum of diameters of connected sets that contain both {a} and {b}. This is indeed a metric if the space {X} is reasonable enough (i.e., any two points are contained in some bounded connected set). On a convex subset of {\mathbb R^n}, the inner diameter metric coincides with the Euclidean metric {d_2}.

One might think that the equality {\rho=d_e} should imply that the domain is convex, but this is not so. Indeed, consider the union of three quadrants on a plane, say {A = \{(x,y) \colon x > 0\text{ or }y > 0\}}. Any two points of {A} can be connected by going up from whichever is lower, and then moving horizontally. The diameter of a right triangle is equal to its hypotenuse, which is the Euclidean distance between the points we started with.

A non-convex domain where inner diameter metric is the same as Euclidean

Inner diameter metric comes up (often implicitly) in complex analysis. By the Riemann mapping theorem, every simply connected domain {G\subset \mathbb C}, other than {\mathbb C} itself, admits a conformal map {f\colon G\to \mathbb D} onto the unit disk {D}. This map need not be uniformly continuous in the Euclidean metric (the slit plane is one example), but it is uniformly continuous with respect to the inner diameter metric on {G}.

Furthermore, by normalizing the situation in a natural way (say, {G \supset \mathbb D} and {f(0)=0}), one can obtain a uniform modulus of continuity for all conformal maps {f} onto the unit disk, whatever the domain is. This uniform modulus of continuity can be taken of the form {\omega(\delta) = C\sqrt{\delta}} for some universal constant {C}. Informally speaking, this means that a slit domain is the worst that can happen to the continuity of a conformal map. This fact isn’t often mentioned in complex analysis books. A proof can be found in the book Conformally Invariant Processes in the Plane by Gregory Lawler, Proposition 3.85. A more elementary proof, with a rougher estimate for the modulus of continuity, is on page 15 of lecture notes by Mario Bonk.

Gromov-Hausdorff convergence

The Hausdorff distance {d_{ H}(A,B)} between two subsets {A,B} of a metric space {X} is defined by {d_{ H}(A,B)=\inf\{r>0: A\subset B_r \text{ and } B\subset A_r \}}, where {A_r,B_r} are (open/closed does not matter) {r}-neighborhoods of the sets. Informally: {d_{ H}(A,B)<r} if no matter where you are in one set, you can jump into the other by traveling less than {r}. For example, the distance between letters S and U is about the length of the longer green arrow.

Hausdorff distance
Hausdorff distance

Generally, one assumes the sets to be closed (to avoid zero distance) and bounded (to avoid infinite distance). But I will not; in this post I’m not really interested in verifying all the axioms of a metric.

The Gromov-Hausdorff distance is defined between metric spaces {X,Y} as follows: it is the infimum of {d_{ H}(f(X),g(Y))} taken over all isometric embeddings {f\colon X\rightarrow Z} and {g\colon Y\rightarrow Z} into some metric space {Z}.

The infimum over all pairs of embeddings into all conceivable metric spaces does not sound like something you would want to compute in practice. Of course, the matter boils down to equipping the abstract union {X\sqcup Y} with pseudometrics that are compatible with the original metrics on {X} and {Y}.

A more directly computable notion of distance (not necessarily a metric) can be given as follows: {\rho_{GH}(X,Y)} is the infimum of all {\epsilon>0} for which there exist two maps {f\colon X\rightarrow Y} and {g\colon Y\rightarrow X} such that:

  1. {d_X(g\circ f(x),x)\le \epsilon} for all {x\in X}
  2. {d_Y(f\circ g(y),y) \le \epsilon} for all {y\in Y}
  3. {|d_Y(f(x_1), f(x_2)) - d_X(x_1,x_2)| \le \epsilon } for all {x_1,x_2\in X}
  4. {|d_X(g(y_1), g(y_2)) - d_Y(y_1,y_2)| \le \epsilon } for all {y_1,y_2\in Y}

This is not as elegant as “infimize over all metric space”, but is more practical. For example, it is easy to check that the sequence of one-sheeted hyperboloids {H_n = \{x^2+y^2=z^2+1/n\}}

One hyperboloid,...
One hyperboloid,…
... another hyperboloid, ...
… another hyperboloid, …

converges to the cone {C = \{x^2+y^2=z^2\}}.

... and we get a cone.
… and we get a cone.

Using cylindrical coordinates, define {f_n\colon H_n\rightarrow C} by {f_n(r,\theta,z) = (\sqrt{r^2-1/n}, \theta,z) } and {g\colon C\rightarrow H_n} by {g_n(r,\theta,z) = (\sqrt{r^2+1/n}, \theta,z)}, with an arbitrary choice of {\theta} at the point {g(0,0,0)}. Now check the items one by one:

  1. {f_n\circ g_n} is the identity map on {C}
  2. {g_n\circ f_n} fixes all points of {H_n} except for those with {z=0}. The latter are displaced by at most {2/\sqrt{n}}.
  3. follows from {|\sqrt{r^2-1/n} - r|\le 1/\sqrt{n}} on {[1/n,\infty)}
  4. follows from {|\sqrt{r^2+1/n} - r|\le 1/\sqrt{n}} on {[0,\infty)}

Note that the Gromov-Hausdorff convergence of manifolds is understood with respect to their intrinsic metrics. Although both {H_n} and {C} are naturally identified with subsets of {\mathbb R^3}, it would be a mistake to use the Hausdorff distance based on the Euclidean metric of {\mathbb R^3}. Even though this extrinsic metric is bi-Lipschitz equivalent to the intrinsic metric on both {H_n} and {C}, bi-Lipschitz equivalence is too coarse to preserve the GH convergence in general. In general, the intrinsic metric on a manifold cannot be realized as the extrinsic metric in any Euclidean space.

In the example of hyperboloids we are lucky to have a Gromov-Hausdorff convergent sequence of unbounded spaces. Normally, bounded sets are required, and boundedness is imposed either by an exhaustion argument, or by changing the metric (moving to the projective space). For instance, the parabolas {y=x^2/n} do not converge to the line {y=0} as unbounded subsets of {\mathbb R^2}, but they do converge as subsets of {\mathbb R\mathbb P^2}. In the process, the degree jumps down from {2} to {1}.

It seems that the Gromov-Hausdorff limit of algebraic varieties of degree at most {d} is also an algebraic variety of degree at most {d} (provided that we use the intrinsic metric; otherwise flat ellipses {x^2+n^2y^2=1} would converge to a line segment). If this is true, I’m sure there’s a proof somewhere but I never saw one.