Rearrangement-invariant norms commuting with dilations

We study rearrangement-invariant spaces $X$ over $[0,\mathrm{\infty })$ for which there exists a function $h:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ such that $$\mathrm{\parallel }{D}_{r}f{\mathrm{\parallel }}_X=h(r)\mathrm{\parallel }f{\mathrm{\parallel }}_X$$ for all $f\in X$ and all $r>0$, where $D_r$ is the dilation operator. It is shown that this may hold only if $h(r)=r^{-\frac{1}{p}}$ for all $r>0$, in which case the norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_X$ is called $p$-homogeneous. We investigate which types of r.i. spaces satisfy this condition and show some important embedding properties.