Subdifferential and properties of convex functions with respect to vector fields
Subdifferential and properties of convex functions with respect to vector fields
复制标题
凸函数关于向量场的次微分和性质
DOI：
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发表时间：
2014
影响因子：
0.6
通讯作者：
F. Dragoni
中科院分区：
文献类型：
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作者：
M. Bardi;F. Dragoni
我们研究函数凸关于一个给定的家庭X的向量场，一个概念，自然出现在Carnot-Carath eodory度量空间的性质。我们定义了一个合适的次微分，并证明了一个连续函数是X -凸的当且仅当这样的次微分在每一点上都是非空的。对于卡诺型向量场，我们从这一性质推出广义Fenchel变换是对合的，是詹森不等式的弱形式。最后，我们介绍并比较了几种X-仿射函数的概念，并说明了它们与X-凸性的关系。
We study properties of functions convex with respect to a given family X of vector fields, a notion that appears natural in Carnot-Carath eodory metric spaces. We define a suitable sub- differential and show that a continuous function is X -convex if and only if such subdifferential is nonempty at every point. For vector fields of Carnot type we deduce from this property that a generalized Fenchel transform is involutive and a weak form of Jensen inequality. Finally we introduce and compare several notions of X-affine functions and show their connections with X-convexity.