A (non-mathematical) book I was reading made me wonder whether there was a well-defined notion of the limit of a sequence of geometric curves or shapes. For example, in the method of exhaustion, the sequence of $n$-gons in a circle have an area that converges to the area of their circumscribing circle, but that's a much weaker property than stating that they actually converge to the circle itself, in some precise way. The devil, of course, is always in the details, and I set out to define a limit for a sequence of curves. However, I came up with a couple reasonable-seeming definitions, and cannot prove that they are equivalent, perhaps under some restriction on the properties of the curves (be it continuity, piecewise smoothness, etc.).
Definition 1: Distance measure
Let $G$ and $H$ be two geometric shapes, where a shape is understood to be a topologically closed subset of $\mathbb{R}^2$. We define the distance of $H$ relative to $G$ by $$ \newcommand{\norm}[1]{\left\lVert#1\right\rVert} d_G(H) = \max_{g \in G} \left( \min_{h \in H} \norm{g-h} \right). $$ Note that $d_G(H) \neq d_H(G)$ in general (for example, if $H \supset G$ in which case $d_G(H) = 0$ but $d_H(G) > 0$). Therefore, we define the symmetric distance between the shapes as $$ \norm{G,H} = \max\left( d_G(H), d_H(G) \right). $$
Definition 2: Limit with respect to distance measure
Let $(G_n)$ be an infinite sequence of geometric shapes and $\mathcal{G}$ another geometric shape. The shape $\mathcal{G}$ is defined to be the limit of the sequence $G_n$ if for any arbitrary real number $\epsilon$, there exists some $N$ such that for any $n \geq N$, $\norm{\mathcal{G},G_n} < \epsilon$. In such case, we write $$ \mathcal{G} = \lim_{n \to \infty} G_n. $$
Uniqueness property
A property that can be immediately proven is that if a sequence of shapes $(G_n)$ converges to a (necessarily closed) shape $\mathcal{G}$, then $\mathcal{G}$ is exactly the set of all points in $\mathbb{R}^2$ that are the limit of some sequence of points $g_n$, where $g_n \in G_n$ for each $n$; I'll denote this set $\widehat G$ for convenience. As a corollary, the limit of a geometric sequence $(G_n)$, if it exists, is unique (as it is equal to the uniquely defined $\widehat G$). I can provide the proof of this if necessary.
Unfortunately, this property is of little value on its own. It is easy to come up with examples of sequences $(G_n)$ that do not converge but do have a non-empty, closed $\widehat G$; e.g. two alternating squares sharing a common edge. The result may be helpful in some cases, where perhaps calculating the shape $\widehat G$ is straightforward and one can learn what the limit is, if it exists; perhaps one can then prove that the sequence converges to $\widehat G$ directly. Still, to be generally useful, this result must be partnered with a general characterization of when a geometric sequence converges. This eludes me. Perhaps an analogy to Cauchy sequences would suffice?
Conjecture: Let $(G_n)$ be a geometric sequence. If for every $\epsilon > 0$ there exists some $N$ such that for all $m,n \geq N$, $\norm{G_m,G_n} < \epsilon$ -- a geometric equivalent of a Cauchy sequence -- then $(G_n)$ converges to some closed set $\mathcal{G}$. (In other words, the metric space of geometric shapes is complete).