# Quermassintegrals as mean curvature integrals

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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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 '12 at 6:56