Let $M$ be a compact body in $R^3$ with sufficiently smooth boundary and let $t$ be a large real parameter. It is well-known that the number $N(M;t)$ of points with integer coordinates in a linearly enlarged copy $tM$ is asymptotically equal to the volume vol$(M)t^3$.
From a computational point of view, this implies that the volume of $M$ can be approximated with $N(M; t)$ cubes of edge length $1/t$, whose centers are all contained in $M$ and form a subset of $1/tZ^3$. It would be most desirable to obtain similar algorithms for the other intrinsic volumes of $M$, namely the surface area and the total mean curvature. However, it can be shown that the surface area and the total mean curvature of this approximation of $M$ with cubes can signicantly differ from the values of $M$.
To overcome this problem of non-convergence, we suggest the study of persistent intrinsic volumes. The main goal of this talk is to outline the theory of the first persistent intrinsic volume of a body $M$ in $R^n$. In the special case of a body in $R^3$, the first intrinsic volume coincides with its total mean curvature.
This is (ongoing) joint work with Herbert Edelsbrunner.