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Every polyhedron can be decomposed into a Minkowski sum (or vector sum) of a bounded polyhedron and a polyhedral cone. This paper establishes similar statements for some classes of discrete sets in discrete convex analysis, such as integrally convex sets, L ♮ -convex sets, and M ♮ -convex sets.

L-convex sets are one of the most fundamental concepts in discrete convex analysis. Furthermore, the Minkowski sum of two L-convex sets, called L 2 -convex sets, is an intriguing object that is closely related to polymatroid intersection. This paper reveals the polyhedral description of an L 2 -convex set, together with the observation that the convex hull of an L 2 -convex set is a box-TDI polyhedron. Two different proofs are given for the polyhedral description. The first is a structural short proof, relying on the conjugacy theorem in discrete convex analysis, and the second is a direct algebraic proof, based on Fourier–Motzkin elimination. The obtained results admit natural graph representations. Implications of the obtained results in discrete convex analysis are also discussed.

Discrete Fenchel duality is one of the central issues in discrete convex analysis. The Fenchel-type min–max theorem for a pair of integer-valued M ♮ -convex functions generalizes the min–max formulas for polymatroid intersection and valuated matroid intersection. In this paper we establish a Fenchel-type min–max formula for a pair of integer-valued integrally convex and separable convex functions. Integrally convex functions constitute a fundamental function class in discrete convex analysis, including both M ♮ -convex functions and L ♮ -convex functions, whereas separable convex functions are characterized as those functions which are both M ♮ -convex and L ♮ -convex. The theorem is proved by revealing a kind of box integrality of subgradients of an integer-valued integrally convex function. The proof is based on the Fourier–Motzkin elimination.

Integrally convex functions constitute a fundamental function class in discrete convex analysis, including M-convex functions, L-convex functions, and many others. This paper aims at a rather comprehensive survey of recent results on integrally convex functions with some new technical results. Topics covered in this paper include characterizations of integral convex sets and functions, operations on integral convex sets and functions, optimality criteria for minimization with a proximity-scaling algorithm, integral biconjugacy, and the discrete Fenchel duality. While the theory of M-convex and L-convex functions has been built upon fundamental results on matroids and submodular functions, developing the theory of integrally convex functions requires more general and basic tools such as the Fourier–Motzkin elimination.

The epsilon-subdifferential of convex univariate piecewise linear-quadratic (PLQ) functions can be computed in linear worst-case time complexity as the level-set of a convex function. Using binary search, we improve the complexity to logarithmic worst-case time, and prove such complexity is optimal. In addition, a new algorithm to compute the entire graph of the epsilon-subdifferential in (optimal) linear time is presented. Both algorithms are not limited to convex PLQ functions but are also applicable to any convex piecewise-defined functions with little restrictions.

M♮ -concave functions form a class of discrete concave functions in discrete convex analysis, and are defined by a certain exchange axiom. We show in this paper that M♮ -concave functions can be characterized by a combination of two simpler exchange properties. It is also shown that for a function defined on an integral polymatroid, a much simpler exchange axiom characterizes M♮ -concavity. These results have some significant implications in discrete convex analysis.

The multiple exchange property for matroid bases is generalized for valuated matroids and M-natural concave set functions. The proof is based on the Fenchel-type duality theorem in discrete convex analysis. The present result has an implication in economics: The strong no complementarities condition of Gul and Stacchetti is, in fact, equivalent to the gross substitutes condition of Kelso and Crawford.

We propose the first algorithm to compute the conjugate of a bivariate Piecewise Linear-Quadratic (PLQ) function in optimal linear worst-case time complexity. The key step is to use a planar graph, called the entity graph, not only to represent the entities (vertex, edge, or face) of the domain of a PLQ function but most importantly to record adjacent entities. We traverse the graph using breadth-first search to compute the conjugate of each entity using graph-matrix calculus, and use the adjacency information to create the output data structure in linear time.

This article finds q- and h-integral inequalities in implicit form for generalized convex functions. We apply the definition of q−h-integrals to establish some new unified inequalities for a class of α,ℏ−m-convex functions. Refinements of these inequalities are given by applying a class of strongly α,ℏ−m-convex functions. Several q-integral inequalities for various kinds of convex and strongly convex functions are deduced under specific conditions.

Let$\\pi$be a${\\rm SL}(3,\\Bbb{Z})$Hecke-Maass cusp form. Let$\\chi=\\chi_1\\chi_2$be a Dirichlet character with$\\chi_i$primitive modulo$M_i$ . Suppose$M_1$ ,$M_2$are primes such that$\\sqrt{M_2}M^{4\\delta}

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