  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 with Dirichlet boundary condition around zero, which is
called boundary concentration phenomena. We first examine relation between
boundary concentration phenomena and large spectral gap phenomena of Dirichlet
eigenvalues of Laplacian. We will obtain analogue of the Gromov-V. D. Milman
theorem and the Funano-Shioya theorem for closed manifolds. Furthermore, to
capture boundary concentration phenomena, we introduce a new invariant called
the observable inscribed radius. We will formulate comparison theorems for such
invariant under a lower Ricci curvature bound, and a lower mean curvature bound
for the boundary. Based on such comparison theorems, we investigate various
boundary concentration phenomena of sequences of manifolds with boundary.
