TITLE: Do Minkowski sums have anything like calculus?
QUESTION [7 upvotes]: Is there anything resembling differential calculus over the space of (nicely behaved) regions in $\mathbb{R}^d$, where addition is interpreted in terms of Minkowski sums?  For example, it is known that Minkowski sums act linearly on the perimeter of two-dimensional convex bodies.  Is there any sense in which one could define something like a (constant) gradient for the perimeter function?  I would assume that if anything, said gradient would just be a circle centered at the origin.

REPLY [6 votes]: There is a theory for convex bodies. If $A, B \subset \mathbb{R}^n$ are convex bodies, whose interiors contain the origin, you can use set addition to define $A+tB$, for any $t \ge 0$, and
$$
V(A,B) = \lim_{t\rightarrow 0+} \frac{A+tB}{t}.
$$
This is known as the first mixed volume. $h_A$ and $h_B$ are the support functions, then
$$
V(A,B) = \int_{S^{n-1}} h_B\,dS_A,
$$
where $S^{n-1}$ is the unit sphere and $dS_A$ is known as the surface area measure of $A$. In this sense, the gradient of the volume function is this measure. If $B$ is the unit ball, then $V(A,B)$ is the surface area (perimeter if $n = 2$). If you differentiate again, you get higher order curvature measures. It turns out that
$V(A+tB)$ is a polynomial of degree $n$, called the Steiner polynomial, and its coefficients are called mixed volumes.