I have $n$ points $p_i$ strictly interior to a rectangle $R$,
and I would like to connect them with a curve $C$ whose curvature is as low as possible.
Let $\kappa_\max(C)$ be the sharpest (largest absolute value) of the curvature of $C$
at any point along $C$.
More specifically, $C$ should:
(a) pass through the points *in any order*;
(b) be simple, i.e., non-self-intersecting;
(c) remain inside $R$;
and (d) have the minimum $\kappa_\max(C)$ over all $C$ satisfying
(a,b,c).

For example, perhaps the curve left below is optimal.
The right curve has lower curvature but strays exterior to $R$.

I am seeking a lower bound on the minimum of $\kappa_\max$,
as a function of the point configuration and its placement within $R$.
I have seen literature bounding curvature *variation*,
and literature focused on *shortest paths* of bounded curvature,
and literature that permits $C$ to self-cross,
but no literature on my specific collection of constraints.
My $n$ is not large, so a solution for a fixed permutation
would still be quite useful.
If anyone can point me to relevant literature, I would appreciate it.
Thanks!

**Addendum**. Here is what I gather must be Anton Petrunin's idea:

And here is Scott Carnahan's improvement to my example, left: