realizing set is *minimal* if it does not contain a strict subset that is a realizing set. Such a minimal realizing set contains only “significant” events.

\[lem:realizingset\] For two polygonal curves $P$ and $Q$ with $m > 1$ and $n \geq 1$, there exists a minimal realizing set.

Let $\mathcal{E}$ denote the non-empty set of concurrent events at the minimal critical value. By definition, the empty set cannot be a realizing set and $\mathcal{E}$ is a realizing set. Hence, $\mathcal{E}$ contains a minimal realizing set.

The following lemma directly implies that a locally correct [Fréchet matching]{} always exists. Informally, it states that curves have a locally correct matching that is “closer” (except in cell $(1,1)$ or $(m,n)$) than the distance of their realizing set. Further, this matching is linear inside every cell. In the remainder, we use realizing set to indicate a minimal realizing set, unless indicated otherwise.

\[lem:specmatching\] If the free space diagram of two polygonal curves $P$ and $Q$ is connected at value $\varepsilon$, then there exists a locally correct [Fréchet matching]{} $\mu = (\sigma,\theta)$ such that $d_\mu(t) \leq \varepsilon$ for all $t$ with $\sigma(t) \geq 1$ or $ \theta(t) \geq 1$, and $\sigma(t) \leq m-1$ or $\theta(t) 