NDLI: Discrete Convexity

Content Provider	Springer Nature Link
Author	Danilov, V. I. Koshevoy, G. A.
Copyright Year	2006
Abstract	In the paper, we explain what subsets of the lattice ℤ $^{n}$ and what functions on the lattice ℤ $^{n}$ could be called convex. The basis of our theory is the following three main postulates of classical convex analysis: concave functions are closed under sums; they are also closed under convolutions; and the superdifferential of a concave function is nonempty at each point of the domain. Interesting (and even dual) classes of discrete concave functions arise if we require either the existence of superdifferentials and closeness under convolutions or the existence of superdifferentials and closeness under sums. The corresponding classes of convex sets are obtained as the affinity domains of such discretely concave functions. The classes of the first type are closed under (Minkowski) sums, and the classes of the second type are closed under intersections. In both classes, the separation theorem holds true. Unimodular sets play an important role in the classification of such classes. The so-called polymatroidal discretely concave functions, most interesting for applications, are related to the unimodular system $$\mathbb{A}_n : = \{ \pm e_i ,e_i - e_j \}$$ . Such functions naturally appear in mathematical economics, in Gelfand-Tzetlin patterns, play an important role for solution of the Horn problem, for describing submodule invariants over discrete valuation rings, and so on. Bibliography: 6 titles.
Starting Page	1418
Ending Page	1421
Page Count	4
File Format	PDF
ISSN	10723374
Journal	Journal of Mathematical Sciences
Volume Number	133
Issue Number	4
e-ISSN	15738795
Language	English
Publisher	Kluwer Academic Publishers-Consultants Bureau
Publisher Date	2006-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Statistics and Probability

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