Subadditive functions and a relaxation of the homogeneity condition of seminorms

Abstract:

We prove that every locally bounded above at a point subadditive function $f:(0,\infty ) \to \mathbb {R}$ such that $f(rt) \leqslant rf(t),\;t > 0$, for some $r \in (0,1)$ has to be linear. Using this we show among others that the homogeneity condition of a seminorm ${\mathbf {p}}$ in a real linear space $X$ can be essentially relaxed to the following condition: there exists an $r \in (0,1)$ such that ${\mathbf {p}}(rx) \leqslant r{\mathbf {p}}(x)$ for all $x \in X$. A new characterization of the ${{\mathbf {L}}^p}$-norm and one-line proofs of Minkowski’s and Höder’s inequalities are also given.