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.).