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pitaly

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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##

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##