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Abstract
We consider tropical hemispaces, defined as tropically convex sets whose complements are also tropically convex, and tropical semispaces, defined as maximal tropically convex sets not containing a given point. We introduce the concept of (P,R)decomposition. This yields (to our knowledge) a new kind of representation of tropically convex sets extending the classical idea of representing convex sets by means of extreme points and rays. We characterize tropical hemispaces as tropically convex sets that admit a (P,R)decomposition of certain kind. In this characterization, with each tropical hemispace we associate a matrix with coefficients in the completed tropical semifield, satisfying an extended rankone condition. Our proof techniques are based on homogenization (lifting a convex set to a cone), and the relation between tropical hemispaces and semispaces.
Original language  English 

Pages (fromto)  131163 
Journal  Linear Algebra and its Applications 
Volume  440 
Early online date  13 Nov 2013 
DOIs  
Publication status  Published  1 Jan 2014 
Keywords
 Tropical convexity
 Abstract convexity
 Maxplus algebra
 Hemispace
 Semispace
 Rankone matrix
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Dive into the research topics of 'Characterization of tropical hemispaces by (P,R)decompositions'. Together they form a unique fingerprint.Projects
 1 Finished

PerronFrobenius Theory and MaxAlgebraic Combinatorics of Nonnegative Matrices
Butkovic, P.
Engineering & Physical Science Research Council
12/03/12 → 11/03/14
Project: Research Councils