14,176 research outputs found
The sandwich theorem
This report contains expository notes about a function that is
popularly known as the Lov\'asz number of a graph~. There are many ways to
define , and the surprising variety of different
characterizations indicates in itself that should be
interesting. But the most interesting property of is probably
the fact that it can be computed efficiently, although it lies ``sandwiched''
between other classic graph numbers whose computation is NP-hard. I~have tried
to make these notes self-contained so that they might serve as an elementary
introduction to the growing literature on Lov\'asz's fascinating function
The asymmetric sandwich theorem
We discuss the asymmetric sandwich theorem, a generalization of the
Hahn-Banach theorem. As applications, we derive various results on the
existence of linear functionals that include bivariate, trivariate and
quadrivariate generalizations of the Fenchel duality theorem. Most of the
results are about affine functions defined on convex subsets of vector spaces,
rather than linear functions defined on vector spaces. We consider both results
that use a simple boundedness hypothesis (as in Rockafellar's version of the
Fenchel duality theorem) and also results that use Baire's theorem (as in the
Robinson-Attouch-Brezis version of the Fenchel duality theorem). This paper
also contains some new results about metrizable topological vector spaces that
are not necessarily locally convex.Comment: 17 page
A Ham Sandwich Theorem for General Measures
The "ham sandwich" theorem has been proven only for measures that are absolutely continuous with respect to Lesbeque measure. We prove a generalized version of the ham sandwich theorem which is applicable to arbitrary finite measures, and we give some sufficient conditions for uniqueness of the hyperplane identified by the theorem
Few Cuts Meet Many Point Sets
We study the problem of how to breakup many point sets in into
smaller parts using a few splitting (shared) hyperplanes. This problem is
related to the classical Ham-Sandwich Theorem. We provide a logarithmic
approximation to the optimal solution using the greedy algorithm for submodular
optimization
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