Surprising differences in abstract convexity spaces

In 1966, Helge Tverberg proved that any set of at least (r -
1)(d + 1) + 1 points in dimension d can be partitioned into r parts so
that the total intersection of the convex hulls of the parts is
nonempty. Tverberg's theorem has motivated extensions into the more
general context of abstract convexity spaces. This year, Noga Alon and
Shakhar Smorodinsky showed that a Tverberg-type theorem holds if the
convex hulls of the points are replaced by unions of boundedly many
convex sets, giving an upperbound on the number of points needed. More
recently, Wenchong Chen, Gennian Ge, Yang Shu, Zhouningxin Wang, and
Zixiang Xu provided an example which shows that the upper bound given
by Alon and Smorodinsky is nearly optimal. CGSWX further show that, if
the convex sets in each union are pairwise disjoint, the above bound
can be improved dramatically. We present these recent findings along
with the methods used and discuss the implications for other abstract
convexity spaces.