Hlawka’s functional inequality on topological groups

Abstract

Let $(X,+)$ be a topological abelian group. We discuss regularity of solutions $f:X\to \mathbb{R}$ of Hlawka’s functional inequality

$$f(x+y)+f(y+z)+f(x+z)\le f(x+y+z)+f\left(x\right)+f\left(y\right)+f\left(z\right),$$ postulated for all $x,y,z\in X$. We study the lower and upper hull of $f$. Moreover, we provide conditions which imply continuity of $f$. We prove, in particular, that if $X$ is generated by any neighborhood of zero, $f$ is continuous at zero, and $f\left(0\right)=0$, then $f$ is continuous on $X$.