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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 sufficiently 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
coefficient.
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 ∈ ℝn, gradients with norms bounded a
priori from above and below.
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