Loopy Thinking

Earlier I posted on the 'backwards orientation' in mathematics, aimed at finding new characterizations or 'origin stories' for familiar objects. Here is an exercise in this kind of math, hopefully amusing.

We'll consider the class of simple, rectifiable closed curves in the plane, that is, non-self-intersecting continuous 'loops' in R^2 to whose segments a definite finite arc-length can always be assigned.

Given two points x, y on such a curve C, let d_C(x, y) denote the distance along the curve from x to y (in whichever direction is shorter). On the other hand, let d(x, y) denote regular Euclidean distance.

We say that a curve respects distances if, for all u, v, x, y on C, we have

d_C(u, v) < d_C(x, y) if and only if d(u, v) < d(x, y).

Now, a circle respects distances in this sense; do any other curves in our class?

Would your answer change if we used a weaker notion of respecting distances? Namely, for all x, y, z in C,

d_C(x, y) < d_C(x, z) iff d(x, y) < d(x, z).


Bonus problem: the analogous question for rectifiable-path-connected, closed plane sets with nonempty interior. Convex bodies respect distances; do any others? (Recall that we previously discussed a very different kind of characterization of convexity.)
It may help to know that between any two points in such a set, there exists a path of minimum length (see Kolmogorov & Fomin's book on functional analysis, Sect. 20, Thm 3.).