TITLE: Walking around Santa Cruz, track around the soccer field
QUESTION [11 upvotes]: I was recently walking around the track at UCSC, and I noticed that the track didn't always curve inward. Sometimes it curved the other way. Compare this (A convex track): http://www.irelandaerialphotography.com/dr_f2_3838.jpg to this (A concave track): http://www.motoyard.com/trackday/images/track-chuckwalla-l.gif . In the latter, there are portions of the track where the outer edge ( the edge bordering the infinite face, if this was a planar graph) is longer than the inner edge (the edge bordering the inner field). In a convex track, the outer edge is always longer than the inner edge. But is it possible to construct a (concave) track such that the inner edge is the same length as the outer edge? If not, how does one prove that it is impossible? The track must have a width greater than zero. (If the width is zero, the inner and outer edge will be one and the same, and the question is meaningless.) As an extension, is it possible to construct a track such that all paths along it are of the same length? A path is defined as the set of all points equidistant from the outside edge all the way around the track and equidistant from the inside edge all the way around the track, though these two distances do not have to be equal.

REPLY [3 votes]: Allow self-intersections.  Take the symmetric figure eight.
That does the trick. And it has the bonus that each half is convex. 
This is what Kloeckner might have been trying to say. 
(  As a kid, we used to drive model
slot cars on these. I imagine you can find such tracks raced by real cars. )