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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 $3$?

share|cite|improve this question . It won't have the simple formula as for a curve, but in general you can use the caustic of the normal mapping. – Charlie Frohman Feb 19 '12 at 12:54

For a surface in 3-dimensional Euclidean space, there is an an analogue of the evolute of a curve but ... in Lorentz-Minkowski space-time, not in 3-dimensional 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. 23-24, 1307-1310 (2010).

The author does not consider the case of dimensions greater than 3.

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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 two-dimensions 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 two-dimensions do not extend well to three-dimensions 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.

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