NB: Print in mm for correct size.
I came across these a while ago through Numberphile: https://www.youtube.com/watch?v=cUCSSJwO3GU
Then I decided to make my own shapes of constant width (I made them back in early 2014, then recently saw them in one of my folders and thought I would upload them with a nice rendering), namely the:
2D Shape – Reuleaux Triangle
3D Solid – Revolved Reuleaux Triangle
You can find out a bit more about them here: https://en.wikipedia.org/wiki/Reuleaux_triangle
I also wanted to mention Mitra Smit who made “Meissner bodies of constant width” which I printed a while back and they’re pretty awesome: http://www.thingiverse.com/thing:649488