On the polarity transform

I shall describe the polarity transform for functions, were it came

from, and some of its properties, especially in comparison with the

well known Legendre transform ("duality" versus "polarity").

Then we shall study its (sub)differential structure, and show that

it may be used to solve new families of first order Hamilton--Jacobi type

equations as well as some second order Monge-Ampere type equations.

The first part is based on joint work with Vitali Milman, and the second

part on joint work with Yanir Rubinstein.