ON A SHAPE DERIVATIVE FORMULA IN THE BRUNN-MINKOWSKI THEORY

We extend a formula for the computation of the shape derivative of an integral cost functional with respect to a class of convex domains, using the so-called support functions and gauge functions to express it. This is a priori a formula in shape optimization theory. However, the result also happens to be an extension of a well-known formula from the Brunn-Minkowski theory of convex bodies.