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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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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