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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.