Question

Suppose that you have highly nonconvex preferences for ice cream and olives, like those given in the text, and that you face prices p1, p2 and have m dollars to spend. List the choices for the optimal consumption bundles.

Answer 1

As we know that you would either consume all the ice creams or all polices, the two choices would be following - x1=m/p1 x2=0 or x1=0, x2=m/p2.

In economics the term non-convexity are referred as the violations of convexity assumption.  

Non-convexity is related to the market failure i.e. difference between supply and demand etc.