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Abstract
We explore the relationship between convex and subharmonic functions on discrete
sets. Our principal concern is to determine the setting in which a convex function is
necessarily subharmonic. We initially consider the primary notions of convexity on
graphs and show that more structure is needed to establish the desired result. To
that end, we consider a notion of convexity defined on lattice-like graphs generated
by normed abelian groups. For this class of graphs, we are able to prove that all
convex functions are subharmonic.
Keywords
convex, subharmonic, discrete, graphs
Mathematical Subject Classification 2010
Primary: 26A51
Secondary: 31C20
Milestones
Received: 1 April 2013
Revised: 21 June 2013
Accepted: 5 July 2013
Published: 16 November 2013
Communicated by Ronald Gould