Abstract

We establish useful upper bounds for the (n−1)-area of a level set ρ⁻¹{r} of a general distance function ρ to an (n−1)-dimensional compact subset C of Rⁿ, in terms of r and the area of C. These bounds nicely complement general isoperimetric inequalities that provide lower bounds for the same area. We allow distance functions induced from asymmetric norms, and prove our results without assuming that C is smooth. Unlike standard upper bounds using Federer’s Coarea Formula, which hold only for some values of r and which become arbitrarily large if we restrict r to be contained in suficiently small intervals, our estimates hold for L¹-almost every r > 0.

Our main result both extends and improves upon an important result of Almgren, Taylor, and Wang. First, our estimates hold for general distance functions. Second, in the case of ordinary distance functions, our estimates are sharper than theirs. Because our estimates hold for L¹-almost every r, we can easily integrate to obtain volume estimates, such as those typically required for Hölder continuity theorems for flows in Rⁿ. Indeed, Almgren, Taylor, and Wang used a weaker inequality to establish their main Hölder continuity theorem for curvature-driven flow of the boundary of a single crystal. In that setting, our estimate would lead to a similar result, but with a better coeficient.

We also establish several general results about asymmetric norms and their associated distance functions to compact sets. For example, the latter are Lipschitz continuous and have, for Lⁿ-almost every x in Rⁿ, gradients with norms bounded a priori from above and below.

Mathematical Subject Classification

Primary: 28A75, 52A40

Secondary: 46B20, 49Q10, 49Q15

Authors