3 added 171 characters in body

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).

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 $s(x)=c$.

1

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:

(If curves with tangent discontinuities are permitted, then already a square has $s(x)=c$.)