Abstract: I will give an informal discussion
of the following observations:
-- non-polytopal spheres are hard to find
-- most spheres are non-polytopal
-- methods to prove that a sphere is non-polytopal
This is derived from works by many authors, including Grünbaum, Perles, Mani, Altshuler & Steinberg, Goodman & Pollack, Kalai, as well as Pfeifle & the speaker.