  Teramoto et al. defined a new measure called the gap ratio that measures the
uniformity of a finite point set sampled from $\cal S$, a bounded subset of
$\mathbb{R}^2$. We generalize this definition of measure over all metric spaces
by appealing to covering and packing radius. The definition of gap ratio needs
only a metric unlike discrepancy, a widely used uniformity measure, that
depends on the notion of a range space and its volume. We also show some
interesting connections of gap ratio to Delaunay triangulation and discrepancy
in the Euclidean plane. The major focus of this work is on solving optimization
related questions about selecting uniform point samples from metric spaces; the
uniformity being measured using gap ratio. We consider discrete spaces like
graph and set of points in the Euclidean space and continuous spaces like the
unit square and path connected spaces. We deduce lower bounds, prove hardness
and approximation hardness results. We show that a general approximation
algorithm framework gives different approximation ratios for different metric
spaces based on the lower bound we deduce. Apart from the above, we show
existence of coresets for sampling uniform points from the Euclidean space --
for both the static and the streaming case. This leads to a $\left( 1+\epsilon
\right)$-approximation algorithm for uniform sampling from the Euclidean space.
