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 Zap. Nauchn. Sem. POMI, 2004, Volume 312, Pages 86–93 (Mi znsl774)

Discrete convexity

V. I. Danilov, G. A. Koshevoy

Central Economics and Mathematics Institute, RAS

Abstract: In the paper we explain what sets and functions on the lattice $\mathbb Z^n$ could be called convex. The basis of our theory is the following three main postulates of the classic convex analysis: concave functions are stable under summation, they are also stable under convolution, and the superdifferential of a concave function is nonempty at each point of the domain. Interesting classes of discrete concave functions (and even dual) arise if we require either the existence of superdifferentials and stability under convolution or the existence of superdifferentials and stability under summation. The corresponding classes of convex sets are obtained as the affinity domains of such discretely concave functions. The first type classes are stable under summation and the second type classes are stable under intersection. In both type 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, the most widespread among applications, are related to the unimodular system $\mathbb A_n:=\{\pm e_i,e_i-e_j\}$. Such functions naturally appear in mathematical economics, play an important role for solution the Horn problem, for describing submodule invariants over rings with discrete valuation, in Gelfand–Tzetlin patterns and so on.

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English version:
Journal of Mathematical Sciences (New York), 2006, 133:4, 1418–1421

Bibliographic databases:

UDC: 512

Citation: V. I. Danilov, G. A. Koshevoy, “Discrete convexity”, Representation theory, dynamical systems. Part XI, Special issue, Zap. Nauchn. Sem. POMI, 312, POMI, St. Petersburg, 2004, 86–93; J. Math. Sci. (N. Y.), 133:4 (2006), 1418–1421

Citation in format AMSBIB
\Bibitem{DanKos04} \by V.~I.~Danilov, G.~A.~Koshevoy \paper Discrete convexity \inbook Representation theory, dynamical systems. Part~XI \bookinfo Special issue \serial Zap. Nauchn. Sem. POMI \yr 2004 \vol 312 \pages 86--93 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl774} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2117884} \zmath{https://zbmath.org/?q=an:1075.52508} \elib{http://elibrary.ru/item.asp?id=9129082} \transl \jour J. Math. Sci. (N. Y.) \yr 2006 \vol 133 \issue 4 \pages 1418--1421 \crossref{https://doi.org/10.1007/s10958-006-0057-2} \elib{http://elibrary.ru/item.asp?id=13514783} 

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This publication is cited in the following articles:
1. G. L. Litvinov, “The Maslov dequantization, idempotent and tropical mathematics: a brief introduction”, J. Math. Sci. (N. Y.), 140:3 (2007), 426–444
2. Adivar M., Fang Sh.-Ch., “Convex Optimization on Mixed Domains”, Journal of Industrial and Management Optimization, 8:1 (2012), 189–227
3. A. M. Chudnov, “Weighing algorithms of classification and identification of situations”, Discrete Math. Appl., 25:2 (2015), 69–81
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