Inference of curvature using tubular neighborhoods

Abstract

Geometric inference deals with the problem of recovering the geometry and topology of a compact subset K of ℝd from an approximation by a finite set P. This problem has seen several important developments in the previous decade. Many of the proposed constructions share a common feature: they estimate the geometry of the underlying compact set K using offsets of P, that is r-sublevel set of the distance function to P. These offset correspond to what is called tubular neighborhoods in differential geometry. First and second-order geometric quantities are encoded in the tube Kr around a manifold. For instance, the classical tube formula asserts that it is possible to estimate the curvature of a compact smooth submanifold K from the volume of its offsets. One can hope that if the finite set P is close to K in the Hausdorff sense, some of this geometric information remains in the offsets of P. In this chapter, we will see how this idea can be used to infer generalized notions of curvature such as Federer’s curvature measures.