  We present new results on a number of fundamental problems about dynamic
geometric data structures:
  1. We describe the first fully dynamic data structures with sublinear
amortized update time for maintaining (i) the number of vertices or the volume
of the convex hull of a 3D point set, (ii) the largest empty circle for a 2D
point set, (iii) the Hausdorff distance between two 2D point sets, (iv) the
discrete 1-center of a 2D point set, (v)the number of maximal (i.e., skyline)
points in a 3D point set. The update times are near $n^{11/12}$ for (i) and
(ii), $n^{7/8}$ for (iii) and (iv), and $n^{2/3}$ for (v). Previously,
sublinear bounds were known only for restricted `semi-online' settings [Chan,
SODA 2002].
  2. We slightly improve previous fully dynamic data structures for answering
extreme point queries for the convex hull of a 3D point set and nearest
neighbor search for a 2D point set. The query time is $O(\log^2n)$, and the
amortized update time is $O(\log^4n)$ instead of $O(\log^5n)$ [Chan, SODA 2006;
Kaplan et al., SODA 2017].
  3. We also improve previous fully dynamic data structures for maintaining the
bichromatic closest pair between two 2D point sets and the diameter of a 2D
point set. The amortized update time is $O(\log^4n)$ instead of $O(\log^7n)$
[Eppstein 1995; Chan, SODA 2006; Kaplan et al., SODA 2017].
