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A Hellytype theorem for hyperplane transversals to wellseparated convex sets

Summary: A Helly­type theorem for hyperplane transversals to well­separated
convex sets
Boris Aronov  ,
Jacob E. Goodman + ,
Richard Pollack # ,
Rephael Wenger §
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.


Source: Aronov, Boris - Department of Computer and Information Science, Polytechnic University


Collections: Computer Technologies and Information Sciences