  This paper considers lossy source coding of $n$-dimensional memoryless
sources and shows an explicit approximation to the minimum source coding rate
required to sustain the probability of exceeding distortion $d$ no greater than
$\epsilon$, which is simpler than known dispersion-based approximations.
  Our approach takes inspiration in the celebrated classical result stating
that the Shannon lower bound to rate-distortion function becomes tight in the
limit $d \to 0$. We formulate an abstract version of the Shannon lower bound
that recovers both the classical Shannon lower bound and the rate-distortion
function itself as special cases. Likewise, we show that a nonasymptotic
version of the abstract Shannon lower bound recovers all previously known
nonasymptotic converses.
  A necessary and sufficient condition for the Shannon lower bound to be
attained exactly is presented. It is demonstrated that whenever that condition
is met, the rate-dispersion function is given simply by the varentropy of the
source. Remarkably, all finite alphabet sources with balanced distortion
measures satisfy that condition in the range of low distortions.
  Most continuous sources violate that condition. Still, we show that lattice
quantizers closely approach the nonasymptotic Shannon lower bound, provided
that the source density is smooth enough and the distortion is low. This
implies that fine multidimensional lattice coverings are nearly optimal in the
rate-distortion sense even at finite $n$. The achievability proof technique is
based on a new bound on the output entropy of lattice quantizers in terms of
the differential entropy of the source, the lattice cell size and a smoothness
parameter of the source density. The technique avoids both the usual random
coding argument and the simplifying assumption of the presence of a dither
signal.
