**O. Aichholzer, F. Aurenhammer, E. Demaine, F. Hurtado, P. Ramos, and
J. Urrutia**

We introduce the notion of -convexity and explore polygons in the plane that
have this property. Polygons which are -convex can be triangulated with
fast yet simple algorithms. However, recognizing them is a 3SUM-hard problem.
We give a characterization of 2-convex polygons, a particularly interesting
class, and show how to recognize them in time. A description of
their shape is given as well, which leads to Erdös-Szekeres type results
regarding subconfigurations of their vertex sets. Finally, we introduce the
concept of generalized geometric permutations, and show that their number can
be exponential in the number of 2-convex objects considered.