To follow up on Dirk's observation in the comments, here is a smoothed
version of a Reuleaux triangle with
$s(x)=c$, as illustrated by the dashed normal chords, which each pass through a corner
of the equilateral triangle, on which are centered both the red and the green arcs:
(If curves with tangent discontinuities are permitted, then already a square has $s(x)=c$.)
Of course, the circle also has $s(x)=c$.
For higher dimensions, see the MO question, "Are there smooth bodies of constant width?"
(the answer is: Yes).
