 
Summary: A Hellytype theorem for hyperplane transversals to wellseparated
convex sets
Boris Aronov ,
Jacob E. Goodman + ,
Richard Pollack # ,
Rephael Wenger §
Abstract
Let S be a family of compact convex sets in R d . Let D(S)
be the largest diameter of any member of S . The fam
ily S is eseparated if, for every 0 < k < d, any k of the
sets can be separated from any other d k of the sets by
a hyperplane more than e=D(S) away from all d of the
sets. We prove that if S is an eseparated family of at
least N(e) compact convex sets in R d and every 2d + 2
members of S are met by a hyperplane, then there is a hy
perplane meeting all the members of S . The number N(e)
depends both on the dimension d and on the separation
parameter e. This is the first Hellytype theorem known
for hyperplane transversals to compact convex sets of ar
bitrary shape in dimension greater than one.
