The Minkowski surface of a convex, compact set $K$ can be defined as (here, I am using Kurt Leichtweiß: Konvexe Mengen, Definition 16.2) $$O(K):=\lim_{\epsilon\rightarrow 0+} \frac{V(K_\epsilon)-V(K)}{\epsilon},$$ where $K_\epsilon:=K+B_\epsilon(0)$.

I am currently trying to geometrically grasp this concept. Under which conditions does the Minkowski surface equal the "normal" surface area of a convex compact set? When the two have different values, how do they differ?

An example in the Leichtweiß book states that Minkowski surface and surface area are equal for convex polytopes. This surprised me because my (apparently wrong) idea was that differences would occur when some regularity conditions on the boundary are not met.

  • 1
    $\begingroup$ 1) See here. 2) I advise you to replace the word "surface" by "area". $\endgroup$
    – Jean Marie
    Nov 8, 2021 at 12:16


You must log in to answer this question.

Browse other questions tagged .