  We derive concentration inequalities for the supremum norm of the difference
between a kernel density estimator (KDE) and its point-wise expectation that
hold uniformly over the selection of the bandwidth and under weaker conditions
on the kernel and the data generating distribution than previously used in the
literature. We first propose a novel concept, called the volume dimension, to
measure the intrinsic dimension of the support of a probability distribution
based on the rates of decay of the probability of vanishing Euclidean balls.
Our bounds depend on the volume dimension and generalize the existing bounds
derived in the literature. In particular, when the data-generating distribution
has a bounded Lebesgue density or is supported on a sufficiently well-behaved
lower-dimensional manifold, our bound recovers the same convergence rate
depending on the intrinsic dimension of the support as ones known in the
literature. At the same time, our results apply to more general cases, such as
the ones of distribution with unbounded densities or supported on a mixture of
manifolds with different dimensions. Analogous bounds are derived for the
derivative of the KDE, of any order. Our results are generally applicable but
are especially useful for problems in geometric inference and topological data
analysis, including level set estimation, density-based clustering, modal
clustering and mode hunting, ridge estimation and persistent homology.
