1. The Minkowski sum of two constant-width convex objects, is another. Scaling and/or rotating one, yields another. 2. Consequently, once you find a few, then there should automatically be an infinite set of them. Also, one can perform a Minkowski sum of one object, plus itself rotated by an amount which is distributed via a "normal" distribution with very small "typical angular width" (infinite Minkowski sum) and in this way obtain a constant-width object which is Cinfinity-smooth, but is arbitrarily near in shape to any starting const-width object. 3. Video: http://www.youtube.com/watch?v=jYf3nOYM_mQ 4. One of the cites that has been mentioned: Bernd Kawohl, Christof Weber: Meissner’s Mysterious Bodies, The Mathematical Intelligencer 33,3 (September 2011) 94-101 http://www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf cites many other papers and mentions some remarkable theorems.