In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative $1$-Lipschitz functions vanishing on the boundary. In order to capture such phenomena, we introduce a new invariant called the observable inscribed radius that measures the difference between such $1$-Lipschitz functions and zero. We examine its basic properties, and formulate a comparison theorem under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. 查看全文>>