(Not such a set)
consider a planar set A with the following property. In every direction, the distance between the two parallel lines that touch A from both sides is the same! Must A be a circle?
You can find the answer here.
Recent Comments
Test Your Intuition (5)
This entry was posted in Convexity, Test your intuition and tagged Test your intuition. Bookmark the permalink.
Combinatorics and more 
A model illustrating these forms (realized as “wheels”) can be found in the mathematikum (http://www.mathematicum.de/) – at least a few years ago when I was there (just in case somebody is interested).
I first encountered these construction in Poul Anderson’s story “The Three-Cornered Wheel”, in which the protagonists need to transfer some heavy stuff a long distance on a primitive planet where the shape of a circle is sacred (and therefore forbidden to use).
Dear Ori,
interesting! I did not know they found their way to the literatiue. They can be found (at least implicitely) in the Challenger disaster report.
here is few other links http://kmoddl.library.cornell.edu/tutorials/02/source http://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/OWR-Kawohl-0907a.pdf
A nice picture of two triples of Reuleaux triangles with an interesting property can be found here: http://konradswanepoel.wordpress.com/2009/09/23/3-x-3-reuleaux-triangles/