凸函数（英文：Convex function）是指函数图形上，任意兩點連成的線段，皆位於圖形的上方的实值函数， [1] 如單變數的二次函数和指数函数。 二階可導的一元函數 f {\displaystyle f} 為凸， 当且仅当 其定義域為凸集，且函數的二階導數 f ″ {\displaystyle f''} 在整個定 ...

An M‑convex function f can be extended to a convex function \( {\overline{f}}\colon\ {\mathbb{R}^{V}} \to {\mathbb{R}\cup\{ + \infty\}} \), and the value of \( \overline{f}(x) \) for x ∊ R V is determined by {f(y): y ∊ Z V, ⌊x⌋ ≤ y ≤ ⌈x⌉. That is, an M‑convex function is an integrally convex function in the sense of .

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.

2003年9月12日 · This paper gives two new characterizations of M- and M ♮-convex functions generalizing Gul and Stacchetti's results on the equivalence among the single improvement condition, the gross substitutes condition and the no complementarities condition for set functions (utility functions on {0,1} vectors) as well as Fujishige and Yang's observation ...

Discrete convex analysis, going back to Murota [Mur03], provides a unified framework for submodular functions and valuated matroids. The main building block are M-convex sets, lattice points in integral generalized permutohedra. They form a generalization of matroids from set systems to sets of integer points.

M-convex functions form a class of well-behaved discrete convex functions. They are defined in terms of an exchange axiom and are characterized as functions obtained by piecing together M-convex sets in a consistent way or as collections of distance functions with some consistency.

2024年3月12日 · Abstract: We unify the study of quotients of matroids, polymatroids, valuated matroids and strong maps of submodular functions in the framework of Murota's discrete convex analysis. As a main result, we compile a list of ten equivalent characterizations of quotients for M-convex sets, generalizing existing formulations for (poly)matroids and ...

In this article, we establish a connection between m−convex functions and strongly m−subharmonic (sh m ) functions and, using the well-known properties of sh

2025年2月24日 · Abstract page for arXiv paper 2502.17655: Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions

2022年1月1日 · In the present paper we consider a new operation defined by a convolution of sections of an M ♮-convex function that transforms the given M ♮-convex function to an M-convex function, which we call a compression of an M ♮-convex function.