## Quermassintegrals as mean curvature integrals

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It is well-known that the quermassintegrals $W_{i}$ of a convex body $K\subset \mathbb{R}^{n}$ can be written as a mean curvature integral $$W_{i}(K)=n^{-1}\int_{\partial K}H_{i-1} d\mathcal{H}^{n-1},$$ where $H_{i-1}$ is the $i$th elementary symmetric polynomial in the $n-1$ principal curvatures. This can for example be found in Schneider's book on convex bodies in (4.2.28) on p.210 for convex bodies whose boundary is a regular $C^{2}$ surface (here regular means all principal curvatures are positive). Other authors allow for a larger class, for example in Santaló's book on integral geometry III.13.6, p.224 only $C^{2}$ is assumed, although it seems to me that he also needed positive principal curvatures in the proof. So, I guess, my question is the following: Is this result true for convex $C^{2}$ surfaces without assuming that the surface is regular? If this is not the case, is there an easy counterexample?

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## 1 Answer

These things hold in amazing generality, but you must dig into bit of geometric measure theory, which is the right language for this. Here is the paper that probably started the industry in this direction:

http://www.ams.org/journals/tran/1959-093-03/S0002-9947-1959-0110078-1/S0002-9947-1959-0110078-1.pdf

There are many works in this direction by Schneider, Fu, and Bernig.

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Dear alvarezpaiva, thank you for your answer! Before I start with Federer's paper, I would be interested to know if the result I asked about is really contained in the paper and following; what makes me somewhat doubtful about this is that Schneider knew about these results and even contributed, but does not comment on this specific representation of $W_{i}$ for a more general class, although he has quite extensive notes in his book. – Sebastian Scholtes Jun 7 at 6:56
"5. The curvature measures. In this section several versions of Steiner's formula are derived by a modification of the classical method of [W]; the main innovation is the use of the algebra A**(£). By means of Steiner's formula the curvature measures corresponding to a set with positive reach are defined, and their basic properties are established. The proofs of the cartesian product formula and of the generalized Gauss-Bonnet Theorem were partly suggested by [H, 6.1.9] and by [A; FEl]." – alvarezpaiva Jun 7 at 9:27
Federer is never an easy read. On the other hand, this is the right language for many things in integral geometry so if you're working on this, you may as well learn it.  – alvarezpaiva Jun 7 at 9:30
Actually I only need a citation for this result and the only one that I could find (and understand immediately (-; ) has a proof that is in my opinion flawed.  I spent some time looking trough Federer's work and could not find anything resembling the mean curvature integral I'm looking for (but maybe if one knows what one is looking for it is quite easy to spot). Still I find it quite curious that Schneider would restrict the surfaces unnecessarily to be regular if he knows that he does not need to. Are you sure that for these surfaces the results of Federer give exactly the formula above for  – Sebastian Scholtes Jun 7 at 14:36
the quermassintegrals? Sorry for being annoying, I appreciate your help! – Sebastian Scholtes Jun 7 at 14:37
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