[Cursuviam]

Sure, you can prove a minimal distance. Think of this problem as a metal chain. Each link of the chain enforces a maximal distance on the nodes it adjoins. The chain is bound on either end to some fixed point.

In an non-constrained case, sure, there's no-minimal link distance. You just have a pile of chain.

However, if you make the chain taut between the two points, say by decreasing the number of nodes in the chain or the maximal distance of links, the minimal distance between nodes then approaches the maximal distance between nodes. When you have a chain constrained to the straight line between the two points, the maximal distance between nodes is equal to the minimal distance between nodes.