Anisotropic Minkowski Content for Countably ${\mathcal{H}}^k$-rectifiable Sets

Abstract

We study anisotropic Minkowski content for lower-dimensional rectifiable sets. First, we prove that, for every convex body \(C\subseteq\mathbb R^n\), the \(k\)-dimensional \(C\)-anisotropic Minkowski content of every compact \(k\)-rectifiable set exists. We show that it is given by an integral involving the \((n-k)\)-dimensional volumes of the projections of \(C\) onto the approximate normal spaces of the set. We then establish the same formula for closed countably \(\mathcal H^k\)-rectifiable sets of finite \(\mathcal H^k\)-measure satisfying an AFP-\(k\)-condition relative to the linear span of \(C\), provided that the condition is witnessed by a finite Radon measure. Finally, we prove that, for a countably \(\mathcal H^k\)-rectifiable set, if the formula holds for one full-dimensional convex body, then it holds for every full-dimensional convex body.