In this post, I want to discuss one of the most famous distances between subsets of $$\mathbb{R}^n$$ – the Hausdorff distance. Based on that, I will also introduce some convergence analysis regarding set-valued functions. Most of the theoretical results here can be found in 1 and 2.

## Distance from point to set

Let $$C$$ be a nonempty subset of $$\mathbb{R}^n$$ and $$x$$ be any point in $$\mathbb{R}^n$$, then the distance from $$x$$ to $$C$$ is defined to be the minimal among the distances between $$x$$ and points in $$C$$. Formally,

$\mathrm{dist}(x \to C) = \min\limits_{c \in C} \|x - c\|,$

where $$\|\cdot\|$$ can be any valid norm. Geometrically, let $$x_c$$ denote the projected point from $$x$$ onto $$C$$, then the distance from $$x$$ to $$C$$ is equal to the distance between $$x$$ and $$x_c$$.

Next, we generalize this distance to sets.

## Distance from set to set

Let $$C_1$$ and $$C_2$$ be two nonempty subsets of $$\mathbb{R}^n$$, then the distance from $$C_1$$ to $$C_2$$ is defined to be the supreme over all the distances from points in $$C_1$$ to $$C_2$$. Formally,

$\mathrm{dist}(C_1 \to C_2) = \sup_{x \in C_1} \mathrm{dist}(x \to C_2).$

Geometrically, if the distance from $$C_1$$ to $$C_2$$ is small, then this means $$C_1$$ is contained in some neighbourhood of $$C_2$$. More specifically, we have the following lemma.

Lemma: $$\mathrm{dist}(C_1 \to C_2) \leq \epsilon$$ if and only if $$C_1 \subseteq C_2 + \mathbb{B}(0, \epsilon)$$, where $$\mathbb{B}(0, \epsilon)$$ is the $$\|\cdot\|$$-ball centered at $$0$$ with radius $$\epsilon$$.

## Distance between sets

The Huasdorff-distance between $$C_1$$ and $$C_2$$ is then the symmetrization of the above ideas. Formally, the Huasdorff-distance between $$C_1$$ and $$C_2$$ is defined as

$\mathrm{dist}(C_1, C_2) = \max\{\mathrm{dist}(C_1 \to C_2), \mathrm{dist}(C_2 \to C_1)\}.$

The next picture is obtained from Wikipedia.

Proposition:

$\mathrm{dist}(C_1, C_2) = \sup_{\|d\| = 1} |\sigma_{C_1}(d) - \sigma_{C_2}(d)|.$

Proof: By the previous lemma, we know that $$\mathrm{dist}(C_1, C_2) \leq \epsilon$$ if and only if $$C_1 \subseteq C_2 + \mathbb{B}(0, \epsilon)$$ and $$C_2 \subseteq C_2 + \mathbb{B}(0, \epsilon)$$ and this means that

$\forall \|d\| = 1,\enspace\sigma_{C_1}(d) \leq \sigma_{C_2}(d) + \epsilon \enspace \text{and} \enspace \sigma_{C_1}(d) \leq \sigma_{C_2}(d) + \epsilon.$

Therefore, we can conclude that

$\mathrm{dist}(C_1, C_2) = \sup_{\|d\| = 1} |\sigma_{C_1}(d) - \sigma_{C_2}(d)|.$

## Set convergence

Before introducing the concept of set convergence, we need the following definitions regarding the inner and outer limits of sets.

Definition Let $$\{C_i\}_{i = 1}^{\infty}$$ be a infinite sequence of convex sets in $$\mathbb{R}^n$$. The outer limit or the limes exterior of $$C_i$$’s is the set of all cluster points of all selections. Specifically,

$\lim\mathrm{ext}_{i \to \infty} C_i = \{c \mid \exists (c_j)_{j \in J \subseteq \mathbb{N}}, \enspace \text{such that}\enspace c_j \in C_j, c_j \to c\}.$

The inner limit or the limes interior of $$C_i$$’s is the set of limits of all convergent selections. Specifically,

$\lim\mathrm{int}_{i \to \infty} C_i = \{c \mid \exists (c_i)_{i \in \mathbb{N}}, \enspace \text{such that}\enspace c_i \in C_i, c_i \to c\}.$

It is clear that we always have

$\lim\mathrm{int}_{i \to \infty} \subseteq \lim\mathrm{ext}_{i \to \infty} C_i.$

When these two sets are equal, the common set is defined to be the limit of $$C_i$$’s. Specifically,

$C = \lim_{i \to \infty} C_i \enspace\text{iff}\enspace C = \lim\mathrm{ext}_{i \to \infty} C_i = \lim\mathrm{int}_{i \to \infty} C_i.$

Relation with Hausdorff-distance In general, convergence with respect to Hausdorff-distance is not equivalent to set convergence as just defined. The next example will illustrate that.

Example Let $$C_i = [0, i]$$ and $$C = \mathbb{R}_{\geq 0}$$, then it is obviously that

$\lim_{i \to \infty} C_i = C.$

Hoever, the Hausdorff-distance between $$C_i$$ and $$C$$ is always $$\infty$$.

Is there is a way to link these two convergence? The answer is yes. We just need an additional boundedness restriction.

Theorem Suppose there is a bounded set $$X \subset \mathbb{R}^n$$ such that $$C_i, C \subseteq X$$, then

$\lim_{i \to \infty} C_i = C \enspace \Leftrightarrow \enspace \mathrm{dist}(C_i, C) \to 0.$

#### References

1. Hiriart-Urruty, Jean-Baptiste, and Claude Lemaréchal. Fundamentals of convex analysis. Springer Science and Business Media, 2012.

2. Rockafellar, R. Tyrrell, and Roger J-B. Wets. Variational analysis. Vol. 317. Springer Science & Business Media, 2009.