Can a small Gaussian perturbation break subadditivity?

Abstract

Given an integer $a\ge 1$, a function $f:\mathbb{R}\to \mathbb{R}$ is said to be $a$-subadditive if $$f(ax+y)\le af(x)+f(y)\text{ for all }x,y\in \mathbb{R}.$$ Of course, $1$-subadditive functions (which correspond to ordinary subadditive functions) are $2$-subadditive. % and $3$-subadditive. Answering a question of Matkowski, we show that there exists a continuous function $f$ satisfying $f(0)=0$ which is $2$-subadditive but not $1$-subadditive. In addition, the same example is not $3$-subadditive, which shows that the sequence of families of continuous $a$-subadditive functions passing through the origin is not increasing with respect to $a$. The construction relies on a perturbation of a given subadditive function with an even Gaussian ring, which will destroy the original subadditivity while keeping the weaker property. Lastly, given a positive rational cone $H\subseteq (0,\infty )$ which is not finitely generated, we prove that there exists a subadditive bijection $f:H\to H$ such that ${\mathrm{liminf}}_{x\to 0}f(x)=0$ and ${\mathrm{limsup}}_{x\to 0}f(x)=1$. This is related an open question of Matkowski and {Ś}wi{\k a}tkowski in [Proc. Amer. Math. Soc. 119 (1993), 187--197].