  In this paper, we propose new techniques for solving geometric optimization
problems involving interpoint distances of a point set in the plane. Given a
set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$,
the distance selection problem is to find the $k$-th smallest interpoint
distance among all pairs of points of $P$. The previously best deterministic
algorithm solves the problem in $O(n^{4/3} \log^2 n)$ time [Katz and Sharir,
SIAM J. Comput. 1997 and SoCG 1993]. In this paper, we improve their algorithm
to $O(n^{4/3} \log n)$ time. Using similar techniques, we also give improved
algorithms on both the two-sided and the one-sided discrete Fr\'{e}chet
distance with shortcuts problem for two point sets in the plane. For the
two-sided problem (resp., one-sided problem), we improve the previous work
[Avraham, Filtser, Kaplan, Katz, and Sharir, ACM Trans. Algorithms 2015 and
SoCG 2014] by a factor of roughly $\log^2(m+n)$ (resp., $(m+n)^{\epsilon}$),
where $m$ and $n$ are the sizes of the two input point sets, respectively.
Other problems whose solutions can be improved by our techniques include the
reverse shortest path problems for unit-disk graphs. Our techniques are quite
general and we believe they will find many other applications in future.
