# What is this form of concavity called?

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• pitaly
In summary, the conversation was about a model that produces a form of concavity that the speaker is not familiar with. They asked if anyone knew the name of this form and if it has been studied before. The definition of this form was given as a differentiable function with a gradient map, and it was referred to as ??-concave. The standard form of concavity was also mentioned. It was noted that there are many generalizations of this concept, making it difficult to determine if the speaker's form is a specific generalization or possibly equivalent to one.
pitaly
TL;DR Summary
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##

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.

## 1. What is concavity?

Concavity is a term used in mathematics and science to describe the shape of a curve or surface. It refers to the degree to which a curve or surface is curved in a particular direction.

## 2. How is concavity measured?

Concavity is measured by calculating the second derivative of a function. The sign of the second derivative indicates whether the function is concave up or concave down.

## 3. What is the difference between concave up and concave down?

Concave up refers to a curve or surface that is curved upwards, resembling a smile. Concave down refers to a curve or surface that is curved downwards, resembling a frown.

## 4. What is the form of concavity called when the curve is concave up?

When a curve is concave up, it is said to have a positive concavity. This means that the second derivative is positive, and the curve is curving upwards.

## 5. What is the form of concavity called when the curve is concave down?

When a curve is concave down, it is said to have a negative concavity. This means that the second derivative is negative, and the curve is curving downwards.

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