Sublinear function

Remarks

Every norm on $X$ is sublinear. For example, $p(x) = \abs{x}$ on $\R$ satisfies $\abs{tx} = t\abs{x}$ for $t > 0$ and $\abs{x + y} \le \abs{x} + \abs{y}$. Every linear functional is sublinear, and so is $p(x_1, x_2) = \max(x_1, x_2)$ on $\R^2$. The map $p(x) = x^2$ on $\R$ is not sublinear: $p(2 \cdot 1) = 4$ but $2\, p(1) = 2$, so clause 1 fails. Clause 1 with $t = 2$ and $x = 0$ gives $p(0) = 2p(0)$, so $p(0) = 0$; hence clause 1 also holds for $t = 0$. A sublinear function $p$ with $p(-x) = p(x)$ for all $x$ is exactly a seminorm.