Let $K$ be a convex body and let $\| \cdot \|_{K}$ be the correspoding Minkowski functional $$\| x \|_{K} = \inf\{\lambda > 0 : x \in \lambda K \}$$

Let us consider the following map $f: K \rightarrow \partial \mathring K$ such that $f(x) = \nabla \| x \|_{K}$ Here $\partial \mathring K$ stands for the polar set, i.e. $$\partial \mathring K = \{ y \in \partial K : \sup_{x \in \partial K}{\langle x, y \rangle} \leq 1 \}$$

It is pointed out that $f$ pretends to be a Gauss map $v_{K}: \partial K \rightarrow S^{n-1}$ that maps the outer unit normal to the boundary to the unit sphere. Are there any easy ways to recover it geometrically?

It looks as if there is a direct relationship between the fact that the subgradients of convex functions are presicely the outer normal vector of supporting hyperplanes of sublevel sets and the statement above, but i can't see any fast ways to figure it out.


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