Subadditive, Positively Homogeneous, and Sublinear Functions

Let $X$ be a vector space and let $p:X\to\mathbb{R}$.

• $p$ is subadditive if

  $$p(x+y)\le p(x)+p(y)\quad\text{for all }x,y\in X.$$

• $p$ is positively homogeneous if

  $$p(\lambda x)=\lambda p(x)\quad\text{for all }x\in X,\ \lambda>0.$$

• $p$ is sublinear if it is both subadditive and positively homogeneous.

Sublinear functions appear as domination bounds in the real Hahn–Banach theorem . A primary source of sublinear functions is the Minkowski gauge of an absorbing convex set .