In an Euclidean linear plane, the evolute of a given curve $C$ with support function $h(t)$ can be regarded as a kind of derivative $C'$ of $C$. Indeed, $C'$ has support function $h'(\pi/2 t)$.
Is there any analogue in dimension $3$?
In an Euclidean linear plane, the evolute of a given curve $C$ with support function $h(t)$ can be regarded as a kind of derivative $C'$ of $C$. Indeed, $C'$ has support function $h'(\pi/2 t)$. Is there any analogue in dimension $3$? 


For a surface in 3dimensional Euclidean space, there is an an analogue of the evolute of a curve but ... in LorentzMinkowski spacetime, not in 3dimensional Euclidean space : Derivation of convex surfaces of $\Bbb R^3$ in Lorentz space and study of their focals. C. R., Math., Acad. Sci. Paris 348, No. 2324, 13071310 (2010). The author does not consider the case of dimensions greater than 3. 


I think that the search for analogues in higher dimensions will be much more subtle because the gradient of the support function of a convex body is more related to the polar of the body than to its evolute, and because only in twodimensions lines are also hyperplanes. Nevertheless, maybe you could try in dimension 4 (or any even dimension) by using the following construction of "normal hyperplanes" (instead normal lines): Consider a convex body $K \subset {\mathbb R}^{2n}$ with smooth boundary $\partial K$. Using the identification of $\mathbb R^{2n}$ and $\mathbb C^n$ we can define the normal hyperplane at a point $z \in \partial K$ as $\sqrt{1} \ T_z \partial K$ or, equivalently, as the hyperplane that contains the normal to $\partial K$ at $z$ and the complex subspace of maximal dimension inside the tangent space $T_z \partial K$. This way you should get a sort of evolute (although I admit I've never seen this studied ...) which, being an envelope of hyperplanes, should be possible to study using the support function. Sometimes results in twodimensions do not extend well to threedimensions because there is a little complex or symplectic geometry hidden in them. Maybe (I'm not sure), this is an example of such a result. 

