  In this paper, we consider fixed edgelength $n$-step random walks in
$\mathbb{R}^d$. We give an explicit construction for the closest closed
equilateral random walk to almost any open equilateral random walk based on the
geometric median, providing a natural map from open polygons to closed polygons
of the same edgelength. Using this, we first prove that a natural
reconfiguration distance to closure converges in distribution to a
Nakagami$(\frac{d}{2},\frac{d}{d-1})$ random variable as $n \rightarrow
\infty$. We then strengthen this to an explicit probabilistic bound on the
distance to closure for a random $n$-gon in any dimension with any collection
of fixed edgelengths $w_i$. Numerical evidence supports the conjecture that our
closure map pushes forward the natural probability measure on open polygons to
something very close to the natural probability measure on closed polygons; if
this is so, we can draw some conclusions about the frequency of local knots in
closed polygons of fixed edgelength.
