For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L$^\natural$-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L$^\natural$-convexity and integral convexity and is independent of global/local discrete midpoint convexity. The new convexity, named DDM-convexity, has nice properties satisfied by L$^\natural$-convexity and global/local discrete midpoint convexity. DDM-convex functions are stable under scaling, satisfy the so-called parallelgram inequality and a proximity theorem with the same small proximity bound as that for L$^{\natural}$-convex functions. Several characterizations of DDM-convexity are given and algorithms for DDM-convex function minimization are developed. We also propose DDM-convexity in continuous variables and give proximity theorems on these functions.

中文翻译：

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有向离散中点凸度

对于连续函数，中点凸性表征凸函数。通过考虑离散形式的中点凸性，研究了几种类型的函数离散凸性，包括积分凸性、L$^\natural$-凸性和全局/局部离散中点凸性。我们提出了一种新型的离散中点凸性，它介于 L$^\natural$-凸性和积分凸性之间，并且独立于全局/局部离散中点凸性。新的凸性称为 DDM-凸性，具有满足 L$^\natural$-凸性和全局/局部离散中点凸性的良好特性。DDM-凸函数在缩放下是稳定的，满足所谓的平行四边形不等式和具有与L$^{\natural}$-凸函数相同的小邻近界的邻近定理。给出了 DDM-凸性的几个特征，并开发了 DDM-凸函数最小化的算法。我们还提出了连续变量中的 DDM 凸性，并给出了这些函数的邻近定理。