Sharp Hausdorff content estimates for accessible boundaries of domains in metric measure spaces of controlled geometry

We give a sharp Hausdorff content estimate for the size of the accessible boundary of any domain in a metric measure space of controlled geometry, i.e., a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality for a fixed 1 ≤ p < ∞. This answers a question posed by Jonas Azzam. In the process, we extend the result to every doubling gauge in metric measure spaces which satisfies a codimension one bound.