$m$-Convex functions and their properties

In this work, the concept of $m$-convex $m-cv$ functions in Euclidean space ${{\mathbb{R}}^{n}}$ is defined. The connection between $m-cv$ functions and $s{{h}_{m}}$-functions is established and based on this, several properties of $m-cv$ functions are proven. In the class of $m-cv (D)\bigcap L_{loc}^{\infty }(D)$ the concept of the Hessian is introduced. Additionally, it is proven that for $m = 1$, $(1-cv)$ function cannot take the values $-\infty$ at the any point in the given domain, and that it is always locally bounded above and below. Furthermore, it is also shown that these functions are continuous.