Nonlocal membrane bending: a reflection, the facts and its relevance.

About forty years ago it was realized that phospholipid membranes, because they are composed of two layers, exhibit particular, and specific mechanical properties. This led to the concept of nonlocal membrane bending, often called area difference elasticity. We present a short history of the development of the concept, followed by arguments for a proper definition of the corresponding elastic constant. The effects of the nonlocal bending energy on vesicle shape are explained. It is demonstrated that lipid vesicles, cells and cellular aggregates exhibit phenomena that can only be described in a complete manner by considering nonlocal bending.