I What is this form of concavity called?

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The discussion centers on identifying a specific form of concavity related to a differentiable function defined on a subset of real numbers. The concept is defined through a gradient map and involves a vector that influences the relationship between function values at two points. The standard form of concavity is established when this vector is set to zero. While the specific generalization in question is not readily identifiable, the existence of numerous concavity generalizations complicates the search for equivalence. Understanding this form of concavity may require further exploration of existing literature on the topic.
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What is this form of concavity called?
I'm working on a model which produces a form of concavity which I'm not familiar with. Does anyone know what this form is called and if it has been studied before?

The definition in its differentiable version reads:

Let ##X\subset \mathbb{R}^{n}##. A differentiable function ##f##, defined on ##X##, with a gradient map ##\nabla f## is called ??-concave, if there exists a vector ##\beta \in \mathbb{R}_{++}^{n}##, such that the following holds for all ##x,y\in X:##

##f\left( y\right) -f\left( x\right) -\nabla f\left( x\right) \cdot \left(y-x\right) \leq \beta \cdot \left( y-x\right)##

The standard form of concvaity is obtained by setting ##\beta=0##
 
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I couldn't find your specific generalization, but there are so many generalizations of the concept, that it is hard to tell whether yours is among them or possibly equivalent to any.
 
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