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Let S be the set of functions from $\mathbb{R}^n$ to $\mathbb{R}$ which can be expressed as a difference between two convex functions. Clearly S is closed under sum and difference, however I also suspect it is closed under product, integration (for functions from $\mathbb{R}$ to $\mathbb{R}$), inversion (for invertible functions from $\mathbb{R}$ to $\mathbb{R}$) and reciprocal (for non zero functions). Specifically I would like to show each of these:

Let $f,g \in S$, then
(1) $f*g \in S$
(2) if $n=1$ then $h(t) = \int_{0}^{t} f(x)dx \in S$
(3) if $f^{-1}$ exists then $f^{-1}\in S$
(4) if $\forall x, f(x)\neq 0$ then $1/f \in S$

However unlike the case of summation this is not as easy and I'm not sure how. Also I am not sure if all of them are even true.

More broadly
I would like to know if there is a name for this function space or if it is equivalent to some other space. What are the properties of this function space? Etc.

Mathew
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1 Answers1

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In some literatures, the difference of two convex functions is called DC function. And it is also called delta-convex functions.

OP can find the first two properties from this paper. The product and integration can be refer to Proposition 2.1. The reciprocal can be found in Corollary of this paper since constant function is also DC function.

For inverse of $h$, if $h(x) = f(x) - g(x)$, where $f, g$ is convex function, then \begin{equation} h^{-1}(y) = f^{-1}(t), \quad f^{-1}(t) = g^{-1}(t-y). \end{equation} The $g^{-1}$ is the inverse of convex function, which must be concave or convex, thus $h^{-1}$ is also a DC function. Here we need $f,g,h$ are all invertible.

Zenan Li
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